Chapter 5.1, Problem 27E

To determine

To match: The given trigonometric expression with its simplest form, from the given options.

Answer to Problem 27E

The simplest form of the expression $\sin x\sec x$ is option (b) $\tan x$ .

Explanation of Solution

Given information:

The given trigonometric expression:

  $\sin x\sec x$

The given options:

  $\csc x$ (b) $\tan x$ (c) ${\sin }^{2}x$ (d) $\sin x\tan x$ (e) ${\sec }^{2}x$ (f) ${\sec }^{2}x+{\tan }^{2}x$

As per problem,

The given trigonometric expression:

  $\sin x\sec x$

Use the identity $\sec x=\frac{1}{\cos x}$ to simplify

  $\begin{array}{l}\sin x\sec x=\sin x\frac{1}{\cos x}\text{ [reciprocal identity ]}\\ \text{ = }\frac{\sin x}{\cos x}\text{ [multiply]}\\ \text{ =}\tan x\text{ [quotient identity]}\end{array}$

Hence, the simplest form of the expression $\sin x\sec x$ is option (b) $\tan x$ .