Exercise 1 Page 532

We are asked to find which of the given expressions are equivalent to $1.$ For that, we are going to analyze and simplify each of the expressions separately.

Expression $1$

We want to know the value of the given expression when simplified. In general, when simplifying trigonometric expressions, we use some of the trigonometric identities. In this case, we will use the following reciprocal identity. Let's substitute this into the expression and simplify. Note that here $\theta =x.$ We found that, when simplified, the expression equals $\tan x,$ which is not always equal to $1.$ Let's take a look at some examples.

$x$	$\tan x$
$\frac{\pi }{6}$	$\frac{\sqrt{3}}{3}$
$\frac{\pi }{4}$	$1$
$\frac{\pi }{3}$	$\sqrt{3}$

This means that the expression is not equivalent to $1.$

Expression $2$

Let's take a look at the expression. To see if it is equivalent to $1,$ we can recall one of the Pythagorean identities. The identity tells us that the expression ${\sin }^2\theta +{\cos }^2\theta$ is constant and equal to $1,$ regardless of the value of $\theta .$ In our expression we have $\theta =x.$ Since the argument does not matter, the expression ${\sin }^{2}x+{\cos }^{2}x$ is also equivalent to $1.$

Expression $3$

We will now move on to the third expression. We want to know the value of the given expression when simplified. We will use the three following identities to simplify the expression.

Negative Angle Identity	Tangent Identity
$\sin (\text{-}\theta )=\text{-}\sin \theta$	$\tan \theta =\frac{\sin \theta }{\cos \theta }$
$\cos (\text{-}\theta )=\cos \theta$

Note that in our expression, the argument is $x,$ not $\theta .$ The expression is equivalent to $1.$

Expression $4$

Let's take a look at the last expression. In order to simplify it, we will use the following cofunction and reciprocal identities.

Cofunction Identity	Reciprocal Identity
$\cos \left(\frac{\pi }{2}-\theta \right)=\sin \theta$	$\csc \theta =\frac{1}{\sin \theta }$

Let's do it! We have that the expression is also equivalent to $1.$