Deck 18: Trigonometric Identities and Equations

Question 1

Change $\sin \theta \sec \theta-\tan \theta \csc \theta$ to an expression containing only sin and $\cos$ .

Answer 1

$\frac{\sin \theta-1}{\cos \theta}$

Question 2

Change $\sin \theta \cot \theta+\cos \theta \tan \theta$ to an expression containing only $\sin$ and $\cos$ .

Answer 2

$\cos \theta+\sin \theta$

Question 3

Change $\frac{\tan \theta}{\sec \theta}$ to an expression containing only $\sin$ and $\cos$ .

Answer 3

$\sin \theta$

Question 4

Change $\frac{\csc \theta}{\cot \theta}$ to an expression containing only $\sin$ and $\cos$ .

Question 5

Change $\frac{2-\cos^{2} \theta}{1+\tan^{2} \theta}$ to an expression containing only $\sin$ .

Question 6

Change $\frac{1-\cos^{2} \theta}{1+\cot^{2} \theta}$ to an expression containing only $\sin$ .

Question 7

Simplify $\cot \theta \sin \theta$ .

Question 8

Simplify $\cos \theta \tan \theta$ .

Question 9

Simplify $\csc \theta \tan \theta$ .

Question 10

Simplify $\csc \theta \cos \theta \cot \theta$ .

Question 11

Simplify $\left(1+\tan^{2} \theta\right)\left(\csc^{2} \theta-1\right)$ .

Question 12

Simplify $\left(\sec^{2} \theta-1\right)\left(\cot^{2} \theta\right)$ .

Question 13

Simplify $\frac{(1+tan^2x) cos x}{csc x}$ .

Question 14

Simplify $\frac{\sin \theta+1}{\tan \theta+\sec \theta}$ .

Question 15

Simplify $\frac{\sin^{2} \theta+\cos^{2} \theta}{\sin^{2} \theta-1}$ .

Question 16

Simplify $\sin x \tan x+\frac{\sin x}{\tan x}$ .

Question 17

Simplify $\frac{1-\sin^{2} \theta}{1-\csc^{2} \theta}$ .

Question 18

Prove the identity $\frac{1}{\sin x}=\cot x \sec x$ .

Question 19

Prove the identity $\tan x+\cot x=\sec x \csc x$ .

Question 20

Prove the identity $\cot \theta(\csc \theta-\cot \theta)=\frac{\cos \theta}{1+\cos \theta}$ .

Question 21

Prove the identity $\frac{\tan^{2} \theta+1}{\sec^{2} \theta-1}=\csc^{2} \theta$ .

Question 22

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

Question 23

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

Question 24

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

Question 25

Expand and simplify $\cos \left(\theta+60^{\circ}\right)$ .

Question 26

Expand and simplify $\cos (\alpha+2 \beta)$ .

Question 27

Expand and simplify $\sin \left(x+45^{\circ}\right)$ .

Question 28

Expand and simplify $\cos (x-\pi / 3)$ .

Question 29

Expand and simplify $\sin \left(\theta+30^{\circ}\right)$ .

Question 30

Expand and simplify $\tan \left(\theta-30^{\circ}\right)$ .

Question 31

Expand and simplify $\tan \left(\theta+45^{\circ}\right)$ .

Question 32

Simplify $\cos (\theta+\pi / 4)+\sin (\theta+\pi / 4)$ .

Question 33

Simplify $\cos 3 x \cos 7 x+\sin 3 x \sin 7 x$ .

Question 34

Simplify $\cos \left(\theta-60^{\circ}\right)+\sin \left(\theta+60^{\circ}\right)$ .

Question 35

Simplify $\cos 5 x \cos 2 x-\sin 5 x \sin 2 x$ .

Question 36

Simplify $\frac{\tan \theta+\tan 50^{\circ}}{1-\tan \theta \tan 50^{\circ}}$ .

Question 37

Simplify $\cos 5 x \cos 2 x+\sin 5 x \sin 2 x$ .

Question 38

Simplify $\frac{\cos \left(\theta+30^{\circ}\right)-\cos \left(\theta-30^{\circ}\right)}{\sin \left(\theta+30^{\circ}\right)+\sin \left(\theta-30^{\circ}\right)}$ .

Question 39

Simplify $\left[\sin \left(30^{\circ}+\theta\right)-\cos \left(\theta+30^{\circ}\right)\right]^{2}$ .

Question 40

Prove the identity $\sin (\theta-3 \pi / 2)=\cos \theta$ .

Question 41

Prove the identity $\tan (\theta-\pi / 3)=-\cot (\theta-\pi / 6)$ .

Question 42

Prove the identity $\sin \left(x+30^{\circ}\right)+\sin \left(30^{\circ}-x\right)=\cos x$ .

Question 43

Prove the identity $\cos \left(x+60^{\circ}\right)+\cos \left(60^{\circ}-x\right)=\cos x$ .

Question 44

Prove the identity $\frac{\sec x-\csc x}{\tan x-\cot x}=\frac{\tan x+\cot x}{\sec x+\csc x}$ .

Question 45

Prove the identity $\frac{1+\tan \theta}{1-\tan \theta}=\tan \left(45^{\circ}+\theta\right)$ .

Question 46

Prove the identity $\frac{\sin \left(60^{\circ}+\theta\right)+\sin \left(\theta-60^{\circ}\right)}{\cos \left(30^{\circ}+\theta\right)-\cos \left(30^{\circ}-\theta\right)}=-1$ .

Question 47

Prove the identity $\frac{\sin \left(45^{\circ}+\theta\right)-\cos \left(45^{\circ}+\theta\right)}{\sin \left(45^{\circ}-\theta\right)+\cos \left(45^{\circ}-\theta\right)}=\tan \theta$ .

Question 48

Prove the identity $\frac{\tan A-\tan B}{\tan A+\tan B}=\frac{\sin (A-B)}{\sin (A+B)}$ .

Question 49

Prove the identity $\frac{\tan A+\tan B}{\tan (A+B)}+\frac{\tan A-\tan B}{\tan (A-B)}=2$ .

Question 50

Express $y=11.4 \sin \omega t+15.7 \cos \omega t$ as a single sine function.

Question 51

Express $y=25.40 \sin \omega t+32.40 \cos \omega t$ as a single sine function.

Question 52

Express $y=5.75 \sin \omega t+8.50 \cos \omega t$ as a single sine function.

Question 53

Express $y=12 \sin \omega t+14 \cos \omega t$ as a single sine function.

Question 54

Simplify $(\sin 2 \theta)(\csc \theta)$ .

Question 55

Simplify $(\cos 2 \theta)(\sec \theta)$ .

Question 56

Simplify $(\cos 2 \theta)(\sin 2 \theta)$ .

Question 57

Simplify $\cos^{2} 2 \theta-\sin^{2} 2 \theta$ .

Question 58

Simplify $(\tan 2 \theta)(\cot \theta)$ .

Question 59

Simplify $\frac{2 \csc^{2} x-1}{1+\cos^{2} x}$ .

Question 60

Simplify $\tan 2 x \cos x$ .

Question 61

Simplify $\cos 2 \theta \csc^{2} \theta-\cot^{2} \theta$ .

Question 62

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

Question 63

Prove that $\sin (2 \theta) \cos (2 \theta)=4 \sin \theta \cos^{3} \theta-2 \sin \theta \cos \theta$ .

Question 64

Prove that $\cos^{4} \theta-\sin^{4} \theta=\cos 2 \theta$ .

Question 65

Prove that $\frac{\tan \theta}{1-\tan \theta}+\frac{\tan \theta}{1+\tan \theta}=\tan 2 \theta$ .

Question 66

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

Question 67

Prove that $\frac{1}{1-\tan \theta}-\frac{1}{1+\tan \theta}=\tan 2 \theta$ .

Question 68

Prove that $\cos 2 \theta=\frac{1-\tan^{2} \theta}{1+\tan^{2} \theta}$ .

Question 69

Prove that $\sin 3 \theta=3 \sin \theta \cos^{2} \theta-\sin^{3} \theta$ .

Question 70

Prove that $\cos 3 \theta=\cos^{3} \theta-3 \sin^{2} \theta \cos \theta$ .

Question 71

If $A$ and $B$ are the two acute angles of a right triangle, show that $2 \csc (2 A)-\cot A=\cot B$ .

Question 72

If $A$ and $B$ are the two acute angles of a right triangle, show that $\frac{\sin (A-B)}{\tan A-\tan B}=\frac{1}{2} \sin 2 A$ .

Question 73

Simplify $\frac{1-\tan^{2} \theta}{1+\tan^{2} \theta}$ .

Question 74

Prove that $\cos \frac{\theta}{2} \sin \frac{\theta}{2}=\frac{\sin \theta}{2}$ .

Question 75

Prove the identity $\tan \frac{\theta}{2}+\cot \theta=\csc \theta$ .

Question 76

Prove the identity $2 \cos^{2} \frac{x}{2}-\cos x=1$ .

Question 77

Prove that $\cos \frac{\theta}{4} \sin \frac{\theta}{4}=\frac{1}{2} \sin \frac{\theta}{2}$ .

Question 78

Prove that $\frac{\cos^{2}\left(\frac{x}{2}\right)}{\sin x}=\frac{\csc x+\cot x}{2}$ .

Question 79

Prove that $\sin \theta=\frac{2 \tan \left(\frac{\theta}{2}\right)}{1+\tan^{2}\left(\frac{\theta}{2}\right)}$ .

Question 80

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