Chapter 5.1, Problem 29E

To determine

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

Answer to Problem 29E

The simplest form of the expression $\frac{{\sec }^{2}x-1}{{\sin }^{2}x}$ is option (e) ${\sec }^{2}x$ .

Explanation of Solution

Given information:

The given trigonometric expression:

  $\frac{{\sec }^{2}x-1}{{\sin }^{2}x}$

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:

  $\frac{{\sec }^{2}x-1}{{\sin }^{2}x}$

Use the identity ${\tan }^{2}x={\sec }^{2}x-1$ before simplifying

  $\begin{array}{l}\frac{{\sec }^{2}x-1}{{\sin }^{2}x}=\frac{{\tan }^{2}x}{{\sin }^{2}x}\text{ [pythagorean identity]}\\ \text{ = }\frac{{\tan }^{2}x}{1}\cdot \frac{1}{{\sin }^{2}x}\text{ [product of fractions]}\\ \text{ =}{\left(\frac{\sin x}{\cos x}\right)}^2\cdot \frac{1}{{\sin }^{2}x}\text{ [quotient identity]}\\ \text{ =}\frac{{\sin }^{2}x}{{\cos }^{2}x}\cdot \frac{1}{{\sin }^{2}x}\text{ [rule of exponents]}\\ \text{ =}\frac{1}{{\cos }^{2}x}\text{ [simplify]}\\ {\text{ = sec}}^{2}x\text{ [reciprocal identity]}\end{array}$

Hence, the simplest form of the expression $\frac{{\sec }^{2}x-1}{{\sin }^{2}x}$ is option (e) ${\sec }^{2}x$ .