Simplifying Trigonometric Expressions

Show your work for each problem.

1.
Which expression is equal to $\cos x \tan x$ ?
1. a) $\sec x$
2. b) $\sin x$
3. c) $\cos^2 x$
4. d) $1$
Answer: $\sin x$
$\cos x \cdot \frac{\sin x}{\cos x} = \sin x$ : the factor $\cos x$ cancels.
2.
Simplify $1 - \sin^2 x$ .
1. a) $\cos^2 x$
2. b) $\tan^2 x$
3. c) $-\cos^2 x$
4. d) $\sec^2 x$
Answer: $\cos^2 x$
Subtract $\sin^2 x$ from both sides of $\sin^2 x + \cos^2 x = 1$ : $1 - \sin^2 x = \cos^2 x$ .
3.
Simplify $\tan x \cot x + 4$ . The result is a number: what is it?
Answer: 5
$\tan x \cot x = \frac{\sin x}{\cos x} \cdot \frac{\cos x}{\sin x} = 1$ , so the expression is $1 + 4 = 5$ .
4.
Simplify $\csc x \tan x$ .
Answer: $\sec x$
1. In sine and cosine: $\csc x \tan x = \frac{1}{\sin x} \cdot \frac{\sin x}{\cos x}$ . Which factor cancels? $\sin x$
2. What is left after cancelling? $\frac{1}{\cos x}$
3. Which single function equals $\frac{1}{\cos x}$ ? $\sec x$
5.
Simplify $\cot x \cdot \sin x \cdot \sec x$ .
Answer: 1
1. Rewrite $\cot x$ in sine and cosine. $\frac{\cos x}{\sin x}$
2. Rewrite $\sec x$ . $\frac{1}{\cos x}$
3. Multiply: $\frac{\cos x}{\sin x} \cdot \sin x \cdot \frac{1}{\cos x} = ?$ 1
6.
A student simplified $\dfrac{\sin x}{\tan x}$ to $\sin^2 x$ . Check this at $x = 60°$ . Then find the correct simplified form.
Answer: $\cos x$
1. Evaluate the original expression: $\dfrac{\sin 60°}{\tan 60°} = \dfrac{\sqrt{3}/2}{\sqrt{3}} = ?$ 0.5
2. Evaluate the student's answer: $\sin^2 60° = \left(\frac{\sqrt{3}}{2}\right)^2 = ?$ 0.75
3. The values differ, so the student is wrong. The correct form is $\cos x$ . What is $\cos 60°$ ? 0.5
7.
Simplify $\dfrac{\sec x}{\csc x}$ .
1. a) $\tan x$
2. b) $\sin x \cos x$
3. c) $\cot x$
4. d) $1$
Answer: $\tan x$
$\dfrac{1/\cos x}{1/\sin x} = \frac{1}{\cos x} \cdot \frac{\sin x}{1} = \frac{\sin x}{\cos x} = \tan x$ .
8.
Simplify $\sec^2 x - \tan^2 x + \csc^2 x - \cot^2 x$ . The result is a number: what is it?
Answer: 2
$\sec^2 x - \tan^2 x = 1$ and $\csc^2 x - \cot^2 x = 1$ , so the sum is $1 + 1 = 2$ .
9.
Simplify $\dfrac{\sin x}{\csc x} + \dfrac{\cos x}{\sec x}$ .
Answer: 1
1. Dividing by $\csc x = \frac{1}{\sin x}$ means multiplying by $\sin x$ . What does $\dfrac{\sin x}{\csc x}$ become? $\sin^2 x$
2. In the same way, what does $\dfrac{\cos x}{\sec x}$ become? $\cos^2 x$
3. What is $\sin^2 x + \cos^2 x$ ? 1
10.
Simplify $\dfrac{1 - \cos^2 x}{1 - \sin^2 x}$ .
1. a) $-1$
2. b) $\cot^2 x$
3. c) $1$
4. d) $\tan^2 x$
Answer: $\tan^2 x$
$\dfrac{1 - \cos^2 x}{1 - \sin^2 x} = \dfrac{\sin^2 x}{\cos^2 x} = \tan^2 x$ .
11.
The expression $\dfrac{4 - 4\cos^2 x}{\sin x}$ simplifies to $k\sin x$ . What is $k$ ?
Answer: 4
$\dfrac{4(1 - \cos^2 x)}{\sin x} = \dfrac{4\sin^2 x}{\sin x} = 4\sin x$ , so $k = 4$ .
12.
Simplify $\dfrac{\sin^2 x}{1 - \cos x}$ , then evaluate the result at $x = 60°$ .
Answer: $1 + \cos x$
1. Replace $\sin^2 x$ using a Pythagorean identity. $1 - \cos^2 x$
2. Factor $1 - \cos^2 x$ as a difference of squares. $(1 - \cos x)(1 + \cos x)$
3. Cancel $1 - \cos x$ : the result is $1 + \cos x$ . What is its value at $x = 60°$ ? 1.5
13.
Simplify $\sin x - \sin x \cos^2 x$ and evaluate it at $x = 30°$ .
Answer: $\sin^3 x$
1. Factor out the common factor. $\sin x(1 - \cos^2 x)$
2. Replace $1 - \cos^2 x$ by $\sin^2 x$ . The expression becomes $\sin^n x$ : what is $n$ ? 3
3. Evaluate $\sin^3 30° = \left(\frac{1}{2}\right)^3$ as a decimal. 0.125
14.
Simplify $\dfrac{\cos x}{1 - \sin x} - \dfrac{\cos x}{1 + \sin x}$ .
1. a) $2\cot x$
2. b) $0$
3. c) $2\tan x$
4. d) $2\sec x$
Answer: $2\tan x$
Numerator: $\cos x(1 + \sin x) - \cos x(1 - \sin x) = 2\sin x\cos x$ . Denominator: $1 - \sin^2 x = \cos^2 x$ . So $\dfrac{2\sin x\cos x}{\cos^2 x} = \dfrac{2\sin x}{\cos x} = 2\tan x$ .
15.
Simplify $\dfrac{1}{1 - \sin x} + \dfrac{1}{1 + \sin x}$ and evaluate the result at $x = 60°$ .
Answer: 8
$\dfrac{(1 + \sin x) + (1 - \sin x)}{1 - \sin^2 x} = \dfrac{2}{\cos^2 x} = 2\sec^2 x$ . At $60°$ : $\cos 60° = \frac{1}{2}$ , so $\sec^2 60° = 4$ and the value is $2 \cdot 4 = 8$ .
16.
Simplify $\dfrac{\tan^2 x}{\sec x + 1}$ , then evaluate the result at $x = 60°$ .
Answer: $\sec x - 1$
1. Replace $\tan^2 x$ using a Pythagorean identity. $\sec^2 x - 1$
2. Factor the numerator $\sec^2 x - 1$ . $(\sec x - 1)(\sec x + 1)$
3. Cancel $\sec x + 1$ : the result is $\sec x - 1$ . Evaluate it at $x = 60°$ . 1
17.
Simplify $(\sin x + \cos x)^2 + (\sin x - \cos x)^2$ .
Answer: 2
1. Expand $(\sin x + \cos x)^2$ . $\sin^2 x + 2\sin x\cos x + \cos^2 x$
2. $(\sin x - \cos x)^2 = \sin^2 x - 2\sin x\cos x + \cos^2 x$ . When you add both expansions, what do the middle terms $2\sin x\cos x$ and $-2\sin x\cos x$ give? 0
3. What is left: $2\sin^2 x + 2\cos^2 x = 2(\sin^2 x + \cos^2 x) = ?$ 2
18.
Simplify $\dfrac{\tan x + \cot x}{\sec x \csc x}$ . The result is a number: what is it?
Answer: 1
$\tan x + \cot x = \dfrac{\sin^2 x + \cos^2 x}{\sin x \cos x} = \dfrac{1}{\sin x \cos x}$ and $\sec x \csc x = \dfrac{1}{\sin x \cos x}$ , so the quotient is $1$ .