Identities & Equations

Problem 1

Express $\tan 35^{\circ}+\tan 55^{\circ}$ using $k$ if $\cos 35^{\circ}=k$ .

See Solution

Problem 2

Prove that $\sin 3\theta + \sin \theta = 4 \sin \theta - 4 \sin^3 \theta$ .

See Solution

Problem 3

Find the equivalent expression for $\sin \left(\frac{\pi}{5}\right)$ from the options: A) $-\cos \left(\frac{\pi}{5}\right)$ B) $-\sin \left(\frac{\pi}{5}\right)$ C) $\cos \left(\frac{3 \pi}{10}\right)$ D) $\sin \left(\frac{7 \pi}{10}\right)$

See Solution

Problem 4

Prove that $\cos 2x \cos \frac{x}{2} - \cos 3x \cos \frac{9x}{2} = \sin 5x \sin \frac{5x}{2}$ .

See Solution

Problem 5

Find the equivalent expression for $\sec \theta - \tan \theta \sin \theta$ .

See Solution

Problem 6

Prove that $\operatorname{Cos}^{2} \theta+\operatorname{Sin}^{2} \theta=1$ . Do all rough work on the answer scripts.

See Solution

Problem 7

Show that $\cos^{2} \theta + \sin^{2} \theta = 1$ .

See Solution

Problem 8

Show that $\cos ^{2} \theta+\sin ^{2} \theta=1$ is true for any angle $\theta$ .

See Solution

Problem 9

Prove the identity: $\frac{1}{\cos x}-\frac{\cos x}{1+\sin x}=\tan x$ using trigonometric identities.

See Solution

Problem 10

Prove the identity: $\frac{1}{\cos x}-\frac{\cos x}{1+\sin x}=\tan x$ .

See Solution

Problem 11

$\sin \alpha=\cos \alpha$ 是 $\cos 2 \alpha=0$ 的哪种条件？ A. 充分不必要 B. 必要不充分 C. 充分必要 D. 既不充分也不必要

See Solution

Problem 12

Prove the identity: $\frac{\sin x}{1+\cos x}+\frac{1+\cos x}{\sin x}=\frac{2}{\sin x}$ .

See Solution

Problem 13

Find $\sin \theta$ given $\csc \theta = \frac{\sqrt{44}}{5}$ . Rationalize denominators if needed.

See Solution

Problem 14

Find the other trigonometric functions of $\theta$ if $\sin \theta=-\frac{\sqrt{3}}{6}$ and $\cos \theta>0$ .

See Solution

Problem 15

Find $\sin \theta$ given $\csc \theta = \frac{\sqrt{20}}{4}$ . Simplify your answer, including any radicals.

See Solution

Problem 16

Find $\sin \theta$ given $\csc \theta = \frac{\sqrt{117}}{6}$ . Simplify your answer.

See Solution

Problem 17

Find $\sin \theta$ given $\csc \theta = \frac{\sqrt{18}}{4}$ . Simplify and rationalize if needed.

See Solution

Problem 18

Find $\sin \theta$ given $\csc \theta = \frac{\sqrt{44}}{2}$ . Simplify your answer.

See Solution

Problem 19

Rewrite $\sin 61^{\circ}$ using its cofunction. What is $\sin 61^{\circ}=$ ? (Provide the answer without the degree symbol.)

See Solution

Problem 20

Rewrite $\tan 74^{\circ}$ using its cofunction. What is $\tan 74^{\circ}=$ ? Simplify your answer without the degree symbol.

See Solution

Problem 21

Solve the equations: a) $\frac{2}{\sin x}-\frac{\sin x}{1+\cos x}=\frac{1+\cos x}{\sin x}$ ; c) $\operatorname{tg} x\left(\operatorname{cotg}^{2} x-1\right)=\operatorname{cotg} x\left(1-\operatorname{tg}^{2} x\right)$ .

See Solution

Problem 22

Find $\cos \theta$ if $\sin \theta=\frac{12}{13}$ and $\theta$ is in quadrant II.

See Solution

Problem 23

Find $\sec \theta$ given that $\cos \theta = \frac{7}{8}$ .

See Solution

Problem 24

Find $\cot \theta$ using the reciprocal identity if $\tan \theta = 6$ .

See Solution

Problem 25

Find $\sin \theta$ given $\csc \theta = \frac{\sqrt{117}}{3}$ . Rationalize denominators if needed.

See Solution

Problem 26

Find $\sin \theta$ given $\csc \theta = \frac{\sqrt{28}}{4}$ . Rationalize denominators if needed.

See Solution

Problem 27

Find $\sin \theta$ if $\csc \theta = 31.25$ .

See Solution

Problem 28

Find $\sin \theta$ using the identity, given that $\csc \theta = 6.4$ .

See Solution

Problem 29

Find $\cot \theta$ if $\tan \theta = 1.25$ using the reciprocal identity.

See Solution

Problem 30

Match the expressions using Quotient and Reciprocal Identities:
1. $-\vee \frac{1}{\csc (\theta)}$
2. $-v \frac{1}{\tan (\theta)}$
3. $-v \frac{\sin (\theta)}{\cos (\theta)}$
4. $-v \frac{1}{\sec (\theta)}$
5. $-\vee \frac{1}{\sin (\theta)}$
6. $-v \frac{1}{\cot (\theta)}$
7. $-v \frac{\cos (\theta)}{\sin (\theta)}$
8. $-v \frac{1}{\cos (\theta)}$
Options: a. $\sin (\theta)$ , b. $\cos (\theta)$ , c. $\tan (\theta)$ , d. $\csc (\theta)$ , e. $\sec (\theta)$ , f. $\cot (\theta)$ .

See Solution

Problem 31

Rewrite $\sin x \cdot \cot x + \cos x$ using $\sin x$ and $\cos x$ , then simplify.

See Solution

Problem 32

Rewrite and simplify $\cos x^{*} \cot x + \sin x$ using $\sin x$ and $\cos x$ .

See Solution

Problem 33

Rewrite $\cos x^{2} \cot x + \sin x$ using $\sin x$ and $\cos x$ , then simplify.

See Solution

Problem 34

Find the exact values of $\sec(60^{\circ})$ and $\csc(60^{\circ})$ using trig function reciprocals. No decimals!

See Solution

Problem 35

Prove that $\frac{1+\sin x}{\cos x}+\frac{\cos x}{1+\sin x} \equiv 2 \sec x$ .

See Solution

Problem 36

Prove that $\frac{1-\cos ^{2} x}{\sec ^{2} x-1} = 1-\sin ^{2} x$ .

See Solution

Problem 37

Show that $\frac{\sin x}{1-\cos^{2} x} = \operatorname{cosec} x$ .

See Solution

Problem 38

Prove that $\sec ^{4} \theta - \sec ^{2} \theta = \tan ^{2} \theta + \tan ^{4} \theta$ .

See Solution

Problem 39

Prove that $\frac{1+\sin x}{1-\sin x} \equiv(\tan x+\sec x)^{2}$ .

See Solution

Problem 40

Prove that $\left(\frac{1+\sin x}{\cos x}\right)^{2}+\left(\frac{1-\sin x}{\cos x}\right)^{2} = 2 \sec ^{2} x+2 \tan ^{2} x$ .

See Solution

Problem 41

In $\triangle Q R S$ , if $\sin R=\cos S$ , what can you conclude about $\angle R$ and $\angle S$ ?

See Solution

Problem 42

Find $\cos \theta$ if $\sin \theta=\frac{4}{5}$ and $\theta$ is in quadrant II. Simplify your answer.

See Solution

Problem 43

Find the remaining trigonometric functions of $\theta$ if $\sin \theta=-\frac{\sqrt{3}}{7}$ in quadrant III.

See Solution

Problem 44

Find the other trigonometric functions of $\theta$ if $\sin \theta=-\frac{\sqrt{3}}{7}$ in quadrant III.

See Solution

Problem 45

Rewrite $\cos \left(90^{\circ}-\theta\right) \cot \theta$ using one of the six trigonometric functions of angle $\theta$ . $\cos \left(90^{\circ}-\theta\right) \cot \theta=$

See Solution

Problem 46

Identify the invalid equation from the options below: A. $\sin \frac{\pi}{3}=\cos \frac{\pi}{6}$ B. $\sin \frac{\pi}{4}=\cos \frac{\pi}{4}$ C. $\csc \frac{\pi}{6}=\cos \frac{\pi}{3}$ D. $\tan \frac{\pi}{4}=\cot \frac{\pi}{4}$

See Solution

Problem 47

Rewrite the expression $\frac{\sin ^{2} \theta+\cot ^{2} \theta+\cos ^{2} \theta}{\csc \theta}$ using basic trig identities.

See Solution

Problem 48

Simplify the expression: $y=\cos ^{2} x+\sin ^{2} x+1$ .

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Problem 49

Simplify:
1. $\frac{\cos 45 \cdot \sin 315+2 \tan 120 \cdot \cos 60}{2 \sin 240 \cdot \cos 300}$
2. Prove: $\frac{\cos x}{1+\sin x}-\frac{1-\sin x}{\cos x}=0$
3. Find $x$ for $5 \sin x+3 \cos x=0$ , $0^{\circ} \leq x \leq 360$
4. Simplify: $\cos (100-x)$

See Solution

Problem 50

Simplify the expression: $\frac{\cos (160-x)}{\tan (90+x) \sin (180+x)}$ using reduction formulas.

See Solution

Problem 51

Simplify $\cot \left(\sin ^{-1}(x)\right)$ using a triangle or trigonometric identity, assuming $x > 0$ .

See Solution

Problem 52

Find the value of the trigonometric expression: $\tan \left[\sec ^{-1}(-5)\right]$ .

See Solution

Problem 53

Find the value of $\cos \left(\cot ^{-1}(10)\right)$ using trigonometric identities.

See Solution

Problem 54

Find the value of $\cot \left(\frac{3 \pi}{4}\right)$ and its equivalent from the options given.

See Solution

Problem 55

Find $x$ where $5 \sin x + 3 \cos x = 0$ for $0^{\circ} \leq x \leq 360^{\circ}$ . Simplify $\cos(180 - x)$ .

See Solution

Problem 56

Express $\tan \left(\beta+25^{\circ}\right)$ using its cofunction for acute angles.

See Solution

Problem 57

Given $\sin 15^{\circ}=c$ , find in terms of $c$ : (i) $\sin 195^{\circ}$ , (ii) $\sin 75^{\circ}$ , (iii) $\cot(-15^{\circ})$ .

See Solution

Problem 58

Find $\sin \theta$ given $\sec \theta=\frac{4}{3}$ and $\tan \theta>0$ . What is $\sin \theta$ ?

See Solution

Problem 59

Find $\sin \theta$ and $\cos \theta$ given $\tan \theta=\frac{7}{\sqrt{15}}$ .

See Solution

Problem 60

Find the exact value of $\csc \frac{\pi}{6}$ using $\sin 30^{\circ}=\frac{1}{2}$ . Options: $\frac{2}{3}$ , $3 \sqrt{3}$ , $\frac{\sqrt{3}}{3}$ , $\frac{2 \sqrt{3}}{3}$ .

See Solution

Problem 61

Find the exact value of $\csc \frac{\pi}{6}$ using $\sin 30^{\circ}=\frac{1}{2}$ . Options: $\frac{2}{3}$ , $3 \sqrt{3}$ , $\frac{\sqrt{3}}{3}$ , $\frac{2 \sqrt{3}}{3}$ .

See Solution

Problem 62

Find the exact value of $\csc \frac{\pi}{6}$ using $\sin 30^{\circ}=\frac{1}{2}$ . Options: $\frac{2}{3}$ , $3 \sqrt{3}$ , $\frac{\sqrt{3}}{3}$ , $\frac{2 \sqrt{3}}{3}$ .

See Solution

Problem 63

Find the exact value of $\tan 20^{\circ}-\frac{\cos 70^{\circ}}{\cos 20^{\circ}}$ without a calculator.

See Solution

Problem 64

Find the exact value of $\sec ^{2} \theta$ given that $\tan ^{2} \theta=3$ . No calculator allowed.

See Solution

Problem 65

Find $\sec \theta$ given $\sin \theta=\frac{\sqrt{7}}{4}$ and $\cos \theta=\frac{3}{4}$ . Options: $\frac{3 \sqrt{7}}{7}$ , $\frac{4 \sqrt{7}}{7}$ , $\frac{\sqrt{7}}{3}$ , $\frac{4}{3}$ .

See Solution

Problem 66

Use $\cos 40^{\circ} \approx 0.77$ to find $\sin 50^{\circ}$ using identities. Round to two decimal places.

See Solution

Problem 67

Find $\tan \theta$ given $\sin \theta=\frac{\sqrt{3}}{2}$ . Choices: $\frac{2 \sqrt{3}}{3}$ , $\sqrt{3}$ , $\frac{\sqrt{3}}{3}$ .

See Solution

Problem 68

Simplify $\frac{\tan ^{2} x+1}{1+\cot ^{2} x}$ and identify any errors in the steps shown. What is the correct result?

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Problem 69

$\sin 19^{\circ} = \cos(71)$ (since $71 = 90 - 19$ ).

See Solution

Problem 70

Rewrite $\sec 77^{\circ}$ using its cofunction. What is the simplified answer? $\sec 77^{\circ}=\square$

See Solution

Problem 71

Rewrite $\cot 18.6^{\circ}$ using its cofunction. What is $\cot 18.6^{\circ}=$ ? (Provide a simplified answer without the degree symbol.)

See Solution

Problem 72

Find an acute angle $\beta$ that satisfies the equation $\sec(4\beta + 24^\circ) = \csc(\beta - 4^\circ)$ .

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Problem 73

Solve for the acute angle $\beta$ in the equation: $\sec(2\beta + 25^{\circ}) = \csc(\beta + 8^{\circ})$ .

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Problem 74

If $a$ and $b$ satisfy $a \sin^2 \theta + a \cos^2 \theta = b$ , find $\frac{b}{a}$ . F. -1 G. 0 H. $\frac{1}{2}$ J. 1 K. 2

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Problem 75

Find $\tan \theta$ given $\sin \theta=\frac{1}{4}$ and $\cos \theta=\frac{\sqrt{15}}{4}$ .

See Solution

Problem 76

Find the exact value of $\cot \left(\frac{\pi}{2}-\theta\right)$ if $\tan \theta=7$ .

See Solution

Problem 77

Find the exact value of $\sec \left(\frac{\pi}{2}-\theta\right)$ if $\tan \theta=2$ . Choices: $\sqrt{5}$ , $\frac{\sqrt{5}}{2}$ , 2, $\frac{1}{2}$ .

See Solution

Problem 78

Find $\cot \theta$ if $\cos \theta=\frac{3 \sqrt{10}}{10}$ .

See Solution

Problem 79

Find the exact value of $\csc \theta$ given $\sin \theta=\frac{1}{4}$ and $\cos \theta=\frac{\sqrt{15}}{4}$ .

See Solution

Problem 80

Find the expression equivalent to $\frac{\tan x-1}{\tan x+1}$ using identities. Options include:
1. $\frac{\cot x}{1-\cot x}$
2. $\frac{\cot x}{1+\cot x}$
3. $\frac{1-\cot x}{1+\cot x}$
4. $\frac{1+\cot x}{1-\cot x}$

See Solution

Problem 81

Find the exact value of $-\frac{\sec 10^{\circ}}{\csc 50^{\circ}}$ using identities. Choices: $-1$ , $1$ , $0$ , undefined.

See Solution

Problem 82

Is the statement true or false? Evaluate if $1+\tan^{2} 30.1^{\circ} = -\sec^{2} 30.1^{\circ}$ .

See Solution

Problem 83

If $\sin \theta=0.4$ , what is $\sin (\theta+\pi)$ ? Options: 0.4, -0.4, -0.6, 0.6.

See Solution

Problem 84

If $\tan \theta=a(a \neq 0)$ , what is $\cot \theta$ using reciprocal identities?

See Solution

Problem 85

Find $\tan \theta$ using $\sin \theta=\frac{6 \sqrt{37}}{37}$ and $\cos \theta=\frac{\sqrt{37}}{37}$ .

See Solution

Problem 86

Find $\cot \theta$ using the values $\sin \theta=-\frac{9}{41}$ and $\cos \theta=-\frac{40}{41}$ .

See Solution

Problem 87

Find $\cos \theta$ given $\sin \theta=\frac{1}{2}$ and $\theta$ in QII. $\cos \theta=\square$

See Solution

Problem 88

Find $\sin \theta$ given $\cos \theta = \frac{4}{5}$ and $\theta$ is in QI. $\sin \theta =$

See Solution

Problem 89

Find $\cos \theta$ given $\sin \theta = -\frac{4}{5}$ and $\theta$ is in QIII. Use the first Pythagorean identity. $\cos \theta =$

See Solution

Problem 90

Find $\sin \theta$ if $\cos \theta = \frac{1}{2}$ and $\theta$ is in quadrant I.

See Solution

Problem 91

Express $\cos \theta$ using only $\sin \theta$ .

See Solution

Problem 92

Rewrite $csc \theta$ using only $\cos \theta$ .

See Solution

Problem 93

Rewrite $\csc \theta \cot \theta$ using $\sin \theta$ and $\cos \theta$ , then simplify if possible.

See Solution

Problem 94

Rewrite $\frac{\sin \theta}{\csc \theta}$ using $\sin \theta$ and $\cos \theta$ , and simplify if possible.

See Solution

Problem 95

Rewrite $\sin \theta \cot \theta + 4 \cos \theta$ using $\sin \theta$ and $\cos \theta$ , then simplify.

See Solution

Problem 96

Express $\sec \theta - \tan \theta \sin \theta$ using $\sin \theta$ and $\cos \theta$ , then simplify.

See Solution

Problem 97

Multiply and simplify: $(1-\sin \theta)(1+\sin \theta)$ .

See Solution

Problem 98

Prove that $\cos \theta \tan \theta = \sin \theta$ is an identity by simplifying the left side.

See Solution

Problem 99

Prove the identity $\sin \theta \sec \theta \cot \theta = 1$ by simplifying the left side using $\sin \theta$ and $\cos \theta$ .

See Solution

Problem 100

Prove that $\frac{\csc \theta}{\cot \theta}=\sec \theta$ by simplifying the left side to match the right side.

See Solution

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Identities & Equations

Problem 1

Express tan⁡35∘+tan⁡55∘ \tan 35^{\circ}+\tan 55^{\circ} tan35∘+tan55∘ using k k k if cos⁡35∘=k \cos 35^{\circ}=k cos35∘=k.

Problem 2

Prove that sin⁡3θ+sin⁡θ=4sin⁡θ−4sin⁡3θ\sin 3\theta + \sin \theta = 4 \sin \theta - 4 \sin^3 \thetasin3θ+sinθ=4sinθ−4sin3θ.

Problem 3

Problem 4

Prove that cos⁡2xcos⁡x2−cos⁡3xcos⁡9x2=sin⁡5xsin⁡5x2\cos 2x \cos \frac{x}{2} - \cos 3x \cos \frac{9x}{2} = \sin 5x \sin \frac{5x}{2}cos2xcos2x​−cos3xcos29x​=sin5xsin25x​.

Problem 5

Find the equivalent expression for sec⁡θ−tan⁡θsin⁡θ\sec \theta - \tan \theta \sin \thetasecθ−tanθsinθ.

Problem 6

Prove that Cos⁡2θ+Sin⁡2θ=1\operatorname{Cos}^{2} \theta+\operatorname{Sin}^{2} \theta=1Cos2θ+Sin2θ=1. Do all rough work on the answer scripts.

Problem 7

Show that cos⁡2θ+sin⁡2θ=1 \cos^{2} \theta + \sin^{2} \theta = 1 cos2θ+sin2θ=1.

Problem 8

Show that cos⁡2θ+sin⁡2θ=1\cos ^{2} \theta+\sin ^{2} \theta=1cos2θ+sin2θ=1 is true for any angle θ\thetaθ.

Problem 9

Prove the identity: 1cos⁡x−cos⁡x1+sin⁡x=tan⁡x\frac{1}{\cos x}-\frac{\cos x}{1+\sin x}=\tan xcosx1​−1+sinxcosx​=tanx using trigonometric identities.

Problem 10

Prove the identity: 1cos⁡x−cos⁡x1+sin⁡x=tan⁡x\frac{1}{\cos x}-\frac{\cos x}{1+\sin x}=\tan xcosx1​−1+sinxcosx​=tanx.

Problem 11

sin⁡α=cos⁡α\sin \alpha=\cos \alphasinα=cosα 是 cos⁡2α=0\cos 2 \alpha=0cos2α=0 的哪种条件？ A. 充分不必要 B. 必要不充分 C. 充分必要 D. 既不充分也不必要

Problem 12

Prove the identity: sin⁡x1+cos⁡x+1+cos⁡xsin⁡x=2sin⁡x\frac{\sin x}{1+\cos x}+\frac{1+\cos x}{\sin x}=\frac{2}{\sin x}1+cosxsinx​+sinx1+cosx​=sinx2​.

Problem 13

Find sin⁡θ\sin \thetasinθ given csc⁡θ=445\csc \theta = \frac{\sqrt{44}}{5}cscθ=544​​. Rationalize denominators if needed.

Problem 14

Find the other trigonometric functions of θ\thetaθ if sin⁡θ=−36\sin \theta=-\frac{\sqrt{3}}{6}sinθ=−63​​ and cos⁡θ>0\cos \theta>0cosθ>0.

Problem 15

Find sin⁡θ\sin \thetasinθ given csc⁡θ=204\csc \theta = \frac{\sqrt{20}}{4}cscθ=420​​. Simplify your answer, including any radicals.

Problem 16

Find sin⁡θ\sin \thetasinθ given csc⁡θ=1176\csc \theta = \frac{\sqrt{117}}{6}cscθ=6117​​. Simplify your answer.

Problem 17

Find sin⁡θ\sin \thetasinθ given csc⁡θ=184\csc \theta = \frac{\sqrt{18}}{4}cscθ=418​​. Simplify and rationalize if needed.

Problem 18

Find sin⁡θ\sin \thetasinθ given csc⁡θ=442\csc \theta = \frac{\sqrt{44}}{2}cscθ=244​​. Simplify your answer.

Problem 19

Rewrite sin⁡61∘\sin 61^{\circ}sin61∘ using its cofunction. What is sin⁡61∘=\sin 61^{\circ}=sin61∘=? (Provide the answer without the degree symbol.)

Problem 20

Rewrite tan⁡74∘\tan 74^{\circ}tan74∘ using its cofunction. What is tan⁡74∘=\tan 74^{\circ}=tan74∘=? Simplify your answer without the degree symbol.

Problem 21

Problem 22

Find cos⁡θ\cos \thetacosθ if sin⁡θ=1213\sin \theta=\frac{12}{13}sinθ=1312​ and θ\thetaθ is in quadrant II.

Problem 23

Find sec⁡θ\sec \thetasecθ given that cos⁡θ=78\cos \theta = \frac{7}{8}cosθ=87​.

Problem 24

Find cot⁡θ\cot \thetacotθ using the reciprocal identity if tan⁡θ=6\tan \theta = 6tanθ=6.

Problem 25

Find sin⁡θ\sin \thetasinθ given csc⁡θ=1173\csc \theta = \frac{\sqrt{117}}{3}cscθ=3117​​. Rationalize denominators if needed.

Problem 26

Find sin⁡θ\sin \thetasinθ given csc⁡θ=284\csc \theta = \frac{\sqrt{28}}{4}cscθ=428​​. Rationalize denominators if needed.

Problem 27

Find sin⁡θ\sin \thetasinθ if csc⁡θ=31.25\csc \theta = 31.25cscθ=31.25.

Problem 28

Find sin⁡θ\sin \thetasinθ using the identity, given that csc⁡θ=6.4\csc \theta = 6.4cscθ=6.4.

Problem 29

Find cot⁡θ\cot \thetacotθ if tan⁡θ=1.25\tan \theta = 1.25tanθ=1.25 using the reciprocal identity.

Problem 30

Problem 31

Rewrite sin⁡x⋅cot⁡x+cos⁡x\sin x \cdot \cot x + \cos xsinx⋅cotx+cosx using sin⁡x\sin xsinx and cos⁡x\cos xcosx, then simplify.

Problem 32

Rewrite and simplify cos⁡x∗cot⁡x+sin⁡x\cos x^{*} \cot x + \sin xcosx∗cotx+sinx using sin⁡x\sin xsinx and cos⁡x\cos xcosx.

Problem 33

Rewrite cos⁡x2cot⁡x+sin⁡x\cos x^{2} \cot x + \sin xcosx2cotx+sinx using sin⁡x\sin xsinx and cos⁡x\cos xcosx, then simplify.

Problem 34

Find the exact values of sec⁡(60∘)\sec(60^{\circ})sec(60∘) and csc⁡(60∘)\csc(60^{\circ})csc(60∘) using trig function reciprocals. No decimals!

Problem 35

Prove that 1+sin⁡xcos⁡x+cos⁡x1+sin⁡x≡2sec⁡x\frac{1+\sin x}{\cos x}+\frac{\cos x}{1+\sin x} \equiv 2 \sec xcosx1+sinx​+1+sinxcosx​≡2secx.

Problem 36

Prove that 1−cos⁡2xsec⁡2x−1=1−sin⁡2x\frac{1-\cos ^{2} x}{\sec ^{2} x-1} = 1-\sin ^{2} xsec2x−11−cos2x​=1−sin2x.

Problem 37

Show that sin⁡x1−cos⁡2x=cosec⁡x\frac{\sin x}{1-\cos^{2} x} = \operatorname{cosec} x1−cos2xsinx​=cosecx.

Problem 38

Prove that sec⁡4θ−sec⁡2θ=tan⁡2θ+tan⁡4θ\sec ^{4} \theta - \sec ^{2} \theta = \tan ^{2} \theta + \tan ^{4} \thetasec4θ−sec2θ=tan2θ+tan4θ.

Problem 39

Prove that 1+sin⁡x1−sin⁡x≡(tan⁡x+sec⁡x)2\frac{1+\sin x}{1-\sin x} \equiv(\tan x+\sec x)^{2}1−sinx1+sinx​≡(tanx+secx)2.

Problem 40

Prove that (1+sin⁡xcos⁡x)2+(1−sin⁡xcos⁡x)2=2sec⁡2x+2tan⁡2x\left(\frac{1+\sin x}{\cos x}\right)^{2}+\left(\frac{1-\sin x}{\cos x}\right)^{2} = 2 \sec ^{2} x+2 \tan ^{2} x(cosx1+sinx​)2+(cosx1−sinx​)2=2sec2x+2tan2x.

Problem 41

In △QRS\triangle Q R S△QRS, if sin⁡R=cos⁡S\sin R=\cos SsinR=cosS, what can you conclude about ∠R\angle R∠R and ∠S\angle S∠S?

Express $\tan 35^{\circ}+\tan 55^{\circ}$ using $k$ if $\cos 35^{\circ}=k$ .

Prove that $\sin 3\theta + \sin \theta = 4 \sin \theta - 4 \sin^3 \theta$ .

Prove that $\cos 2x \cos \frac{x}{2} - \cos 3x \cos \frac{9x}{2} = \sin 5x \sin \frac{5x}{2}$ .

Find the equivalent expression for $\sec \theta - \tan \theta \sin \theta$ .

Prove that $\operatorname{Cos}^{2} \theta+\operatorname{Sin}^{2} \theta=1$ . Do all rough work on the answer scripts.

Show that $\cos^{2} \theta + \sin^{2} \theta = 1$ .

Show that $\cos ^{2} \theta+\sin ^{2} \theta=1$ is true for any angle $\theta$ .

Prove the identity: $\frac{1}{\cos x}-\frac{\cos x}{1+\sin x}=\tan x$ using trigonometric identities.

Prove the identity: $\frac{1}{\cos x}-\frac{\cos x}{1+\sin x}=\tan x$ .

$\sin \alpha=\cos \alpha$ 是 $\cos 2 \alpha=0$ 的哪种条件？ A. 充分不必要 B. 必要不充分 C. 充分必要 D. 既不充分也不必要

Prove the identity: $\frac{\sin x}{1+\cos x}+\frac{1+\cos x}{\sin x}=\frac{2}{\sin x}$ .

Find $\sin \theta$ given $\csc \theta = \frac{\sqrt{44}}{5}$ . Rationalize denominators if needed.

Find the other trigonometric functions of $\theta$ if $\sin \theta=-\frac{\sqrt{3}}{6}$ and $\cos \theta>0$ .

Find $\sin \theta$ given $\csc \theta = \frac{\sqrt{20}}{4}$ . Simplify your answer, including any radicals.

Find $\sin \theta$ given $\csc \theta = \frac{\sqrt{117}}{6}$ . Simplify your answer.

Find $\sin \theta$ given $\csc \theta = \frac{\sqrt{18}}{4}$ . Simplify and rationalize if needed.

Find $\sin \theta$ given $\csc \theta = \frac{\sqrt{44}}{2}$ . Simplify your answer.

Rewrite $\sin 61^{\circ}$ using its cofunction. What is $\sin 61^{\circ}=$ ? (Provide the answer without the degree symbol.)

Rewrite $\tan 74^{\circ}$ using its cofunction. What is $\tan 74^{\circ}=$ ? Simplify your answer without the degree symbol.

Find $\cos \theta$ if $\sin \theta=\frac{12}{13}$ and $\theta$ is in quadrant II.

Find $\sec \theta$ given that $\cos \theta = \frac{7}{8}$ .

Find $\cot \theta$ using the reciprocal identity if $\tan \theta = 6$ .

Find $\sin \theta$ given $\csc \theta = \frac{\sqrt{117}}{3}$ . Rationalize denominators if needed.

Find $\sin \theta$ given $\csc \theta = \frac{\sqrt{28}}{4}$ . Rationalize denominators if needed.

Find $\sin \theta$ if $\csc \theta = 31.25$ .

Find $\sin \theta$ using the identity, given that $\csc \theta = 6.4$ .

Find $\cot \theta$ if $\tan \theta = 1.25$ using the reciprocal identity.

Rewrite $\sin x \cdot \cot x + \cos x$ using $\sin x$ and $\cos x$ , then simplify.

Rewrite and simplify $\cos x^{*} \cot x + \sin x$ using $\sin x$ and $\cos x$ .

Rewrite $\cos x^{2} \cot x + \sin x$ using $\sin x$ and $\cos x$ , then simplify.

Find the exact values of $\sec(60^{\circ})$ and $\csc(60^{\circ})$ using trig function reciprocals. No decimals!

Prove that $\frac{1+\sin x}{\cos x}+\frac{\cos x}{1+\sin x} \equiv 2 \sec x$ .

Prove that $\frac{1-\cos ^{2} x}{\sec ^{2} x-1} = 1-\sin ^{2} x$ .

Show that $\frac{\sin x}{1-\cos^{2} x} = \operatorname{cosec} x$ .

Prove that $\sec ^{4} \theta - \sec ^{2} \theta = \tan ^{2} \theta + \tan ^{4} \theta$ .

Prove that $\frac{1+\sin x}{1-\sin x} \equiv(\tan x+\sec x)^{2}$ .

Prove that $\left(\frac{1+\sin x}{\cos x}\right)^{2}+\left(\frac{1-\sin x}{\cos x}\right)^{2} = 2 \sec ^{2} x+2 \tan ^{2} x$ .

In $\triangle Q R S$ , if $\sin R=\cos S$ , what can you conclude about $\angle R$ and $\angle S$ ?

Find $\cos \theta$ if $\sin \theta=\frac{4}{5}$ and $\theta$ is in quadrant II. Simplify your answer.

Find the remaining trigonometric functions of $\theta$ if $\sin \theta=-\frac{\sqrt{3}}{7}$ in quadrant III.

Find the other trigonometric functions of $\theta$ if $\sin \theta=-\frac{\sqrt{3}}{7}$ in quadrant III.

Rewrite $\cos \left(90^{\circ}-\theta\right) \cot \theta$ using one of the six trigonometric functions of angle $\theta$ . $\cos \left(90^{\circ}-\theta\right) \cot \theta=$

Rewrite the expression $\frac{\sin ^{2} \theta+\cot ^{2} \theta+\cos ^{2} \theta}{\csc \theta}$ using basic trig identities.

Simplify the expression: $y=\cos ^{2} x+\sin ^{2} x+1$ .

Simplify the expression: $\frac{\cos (160-x)}{\tan (90+x) \sin (180+x)}$ using reduction formulas.

Simplify $\cot \left(\sin ^{-1}(x)\right)$ using a triangle or trigonometric identity, assuming $x > 0$ .

Find the value of the trigonometric expression: $\tan \left[\sec ^{-1}(-5)\right]$ .

Find the value of $\cos \left(\cot ^{-1}(10)\right)$ using trigonometric identities.

Find the value of $\cot \left(\frac{3 \pi}{4}\right)$ and its equivalent from the options given.

Find $x$ where $5 \sin x + 3 \cos x = 0$ for $0^{\circ} \leq x \leq 360^{\circ}$ . Simplify $\cos(180 - x)$ .

Express $\tan \left(\beta+25^{\circ}\right)$ using its cofunction for acute angles.

Given $\sin 15^{\circ}=c$ , find in terms of $c$ : (i) $\sin 195^{\circ}$ , (ii) $\sin 75^{\circ}$ , (iii) $\cot(-15^{\circ})$ .

Find $\sin \theta$ given $\sec \theta=\frac{4}{3}$ and $\tan \theta>0$ . What is $\sin \theta$ ?

Find $\sin \theta$ and $\cos \theta$ given $\tan \theta=\frac{7}{\sqrt{15}}$ .

Find the exact value of $\csc \frac{\pi}{6}$ using $\sin 30^{\circ}=\frac{1}{2}$ . Options: $\frac{2}{3}$ , $3 \sqrt{3}$ , $\frac{\sqrt{3}}{3}$ , $\frac{2 \sqrt{3}}{3}$ .

Find the exact value of $\csc \frac{\pi}{6}$ using $\sin 30^{\circ}=\frac{1}{2}$ . Options: $\frac{2}{3}$ , $3 \sqrt{3}$ , $\frac{\sqrt{3}}{3}$ , $\frac{2 \sqrt{3}}{3}$ .

Find the exact value of $\csc \frac{\pi}{6}$ using $\sin 30^{\circ}=\frac{1}{2}$ . Options: $\frac{2}{3}$ , $3 \sqrt{3}$ , $\frac{\sqrt{3}}{3}$ , $\frac{2 \sqrt{3}}{3}$ .

Find the exact value of $\tan 20^{\circ}-\frac{\cos 70^{\circ}}{\cos 20^{\circ}}$ without a calculator.

Find the exact value of $\sec ^{2} \theta$ given that $\tan ^{2} \theta=3$ . No calculator allowed.

Find $\sec \theta$ given $\sin \theta=\frac{\sqrt{7}}{4}$ and $\cos \theta=\frac{3}{4}$ . Options: $\frac{3 \sqrt{7}}{7}$ , $\frac{4 \sqrt{7}}{7}$ , $\frac{\sqrt{7}}{3}$ , $\frac{4}{3}$ .

Use $\cos 40^{\circ} \approx 0.77$ to find $\sin 50^{\circ}$ using identities. Round to two decimal places.

Find $\tan \theta$ given $\sin \theta=\frac{\sqrt{3}}{2}$ . Choices: $\frac{2 \sqrt{3}}{3}$ , $\sqrt{3}$ , $\frac{\sqrt{3}}{3}$ .

Simplify $\frac{\tan ^{2} x+1}{1+\cot ^{2} x}$ and identify any errors in the steps shown. What is the correct result?

$\sin 19^{\circ} = \cos(71)$ (since $71 = 90 - 19$ ).

Rewrite $\sec 77^{\circ}$ using its cofunction. What is the simplified answer? $\sec 77^{\circ}=\square$

Rewrite $\cot 18.6^{\circ}$ using its cofunction. What is $\cot 18.6^{\circ}=$ ? (Provide a simplified answer without the degree symbol.)

Find an acute angle $\beta$ that satisfies the equation $\sec(4\beta + 24^\circ) = \csc(\beta - 4^\circ)$ .

Solve for the acute angle $\beta$ in the equation: $\sec(2\beta + 25^{\circ}) = \csc(\beta + 8^{\circ})$ .

If $a$ and $b$ satisfy $a \sin^2 \theta + a \cos^2 \theta = b$ , find $\frac{b}{a}$ . F. -1 G. 0 H. $\frac{1}{2}$ J. 1 K. 2