Interaction with Square Root and Absolute Value Functions

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These problems begin including the square root and absolute value operations to ensure that factoring expressions does not alter the value of the expression.

1 : Which of the following is equal to $\sin ^2 x - \sin ^4 x$ ?

$\cos ^2 x$ $\cos ^2 x - \cos ^4 x$ $\cos ^4 x$ $\cos ^4 x - \cos ^2 x$

2 : Which of the following is equal to $\cos x + \sin x \tan x$ ?

$1 + \sec x$ $\sin ^2 \left (x-\frac {\pi }{2}\right )$ $1+ \cos x$ $\tan x \csc x$

3 : Which of the following is equal to $\sqrt {\sin x} \cos x - \sqrt {\sin ^5 x} \cos x$ ?

$\cos ^3 x \sqrt {\sin x}$ $\cos ^2 x \sqrt {\sin x}$ $\sqrt {1-\sin x} \cos ^x$ $\sqrt {\cos x} \sin ^2 x$

3.1 : Presumably, in doing that problem, you simplified $\sqrt {\sin ^4 x}$ as $\sin ^2 x$ instead of $|\sin ^2 x|$ . Why do we not use the absolute value bars here?

There was no reason for them in the first place. We’re not concerned with the cases where $\sin ^2 x$ might be negative, so they don’t have any effect. $\sin ^2 x$ is either positive or zero for all values of $x$ , so they don’t have any effect. Although they have an effect, the identity remains true without them.

4 : Which of the following is equal to $\frac {\cos x \cot x}{1 - \sin x} - 1$ ?

$1$ $\frac {1- \cos x}{\sin x}$ $\cos ^2 x - \sin ^2 x$ $\csc x$

5 : Which of the following is equal to $(1+ \cos x)(1 + \cos (-x))$ ?

$1$ $\sin ^2 x$ $2 + 2\cos x - \sin ^2 x$ $1 + \cos ^2 x$

6 : Which one of the following simplifies $\sqrt {\frac {1+\sin x}{1-\sin x}}$ using as few absolute value bars as necessary?

$\frac {|1+\sin x|}{|\cos x|}$ $\frac {|1+\sin x|}{\cos x}$ $\frac {1+\sin x}{|\cos x|}$ $\frac {1+\sin x}{\cos x}$

7 : Determine whether each equality is true or false.

(a): $\frac {1}{1-\cos x} - \frac {\cos x}{1+\cos x} = 2\csc ^2 x -1$
True False
(b): $2- \sin ^2 x - \cos ^2 x = \cos ^2 x +1$
True False
(c): $\cos (-x) (\sec (-x) - \tan (-x)) = \sin x +1$
True False
(d): $\cos x - \cos ^3 x = \cos x \sin ^2 x$
True False
(e): $(\cos x +\sin x)^2 = 1$
True False
(f): $\frac {1+\sin x}{\cos x} = \frac {\cos x}{1+\sin x}$
True False
(g): $\left (\frac {\sec ^2 x - \tan ^2 x}{\tan x}\right )\left (\frac {2+2\tan x}{2+2\cot x} \right ) -2\sin ^2 x = \cos ^2 x - \sin ^2 x$
True False