Proving Trigonometric Identities

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Show your work for each problem.

1.
Which statement is an identity (true for every allowed angle $x$ )?
1. a) $\sin^2 x + \cos^2 x = 1$
2. b) $\sin x = \cos x$
3. c) $\sin x + \cos x = 1$
4. d) $\tan x = 1$
2.
Check the identity $(1 + \tan^2 x)\cos^2 x = 1$ at $x = 60°$ . What is the value of the left side? Use $\tan 60° = \sqrt{3}$ and $\cos 60° = \frac{1}{2}$ .
3.
Check numerically that $\tan x \cos x = \sin x$ holds at $x = 30°$ .
4.
Which move is NOT allowed when you prove an identity?
1. a)Rewrite $\tan x$ as $\frac{\sin x}{\cos x}$ on one side
2. b)Cancel a common factor inside a fraction
3. c)Divide both sides by $\cos x$
4. d)Multiply one side by $\frac{1 + \cos x}{1 + \cos x}$
5.
Show that $\sin(2x) = 2\sin x$ is NOT an identity: compute left side minus right side at $x = 90°$ .
6.
Prove that $\sin x \cot x = \cos x$ .
7.
What is the best first step to prove $\dfrac{\sin x}{1 + \cos x} = \dfrac{1 - \cos x}{\sin x}$ , starting from the left side?
1. a)Square both sides
2. b)Add $\cos x$ to both sides
3. c)Multiply numerator and denominator by $1 - \cos x$
4. d)Multiply numerator and denominator by $1 + \cos x$
8.
Evaluate the right side of $\sec x - \cos x = \sin x \tan x$ at $x = 60°$ . Use $\sin 60° = \frac{\sqrt{3}}{2}$ and $\tan 60° = \sqrt{3}$ .
9.
Prove that $\cos x\,(\sec x - \cos x) = \sin^2 x$ .
10.
Prove that $\dfrac{1}{1 - \cos x} + \dfrac{1}{1 + \cos x} = 2\csc^2 x$ .
11.
Which angle is a counterexample that shows $\sin x + \cos x = 1$ is not an identity?
1. a) $360°$
2. b) $90°$
3. c) $45°$
4. d) $0°$
12.
Evaluate $\dfrac{1 - \sin^2 x}{1 - \cos^2 x}$ at $x = 30°$ .
13.
Disprove the claim $(\sin x + \cos x)^2 = 1$ and find what the left side really equals.
14.
Which expression is $\dfrac{1}{1 - \sin x} - \dfrac{1}{1 + \sin x}$ equal to?
1. a) $2\tan x \sec x$
2. b) $2\sin x$
3. c) $2\sec^2 x$
4. d) $2\tan^2 x$
15.
$\sec^4 x - \tan^4 x = 1 + A\tan^2 x$ is an identity. What is the number $A$ ?
16.
Prove that $\dfrac{\cos x}{1 + \sin x} = \dfrac{1 - \sin x}{\cos x}$ .
17.
Prove that $\cot^2 x - \cos^2 x = \cot^2 x \cos^2 x$ .
18.
Prove that $\sin^4 x - \cos^4 x = 1 - 2\cos^2 x$ .