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 ${\sec }^{4}x-{\tan }^{4}x$ is option (f) ${\sec }^{2}x+{\tan }^{2}x$ .

Explanation of Solution

Given information:

The given trigonometric expression:

  ${\sec }^{4}x-{\tan }^{4}x$

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:

  ${\sec }^{4}x-{\tan }^{4}x$

Simplify by using Pythagorean identity ${\tan }^{2}x={\sec }^{2}x-1$ after factorizing.

  $\begin{array}{l}{\sec }^{4}x-{\tan }^{4}x=({\sec }^{2}x+{\tan }^{2}x)({\sec }^{2}x-{\tan }^{2}x)\text{ [factorise ]}\\ \text{ = }({\sec }^{2}x+{\tan }^{2}x)({\sec }^{2}x-({\sec }^{2}x-1))\text{ [pythagorean identity]}\\ \text{ = }({\sec }^{2}x+{\tan }^{2}x)({\sec }^{2}x-{\sec }^{2}x+1)\text{ [multiply]}\\ \text{ = }({\sec }^{2}x+{\tan }^{2}x)(1)\text{ [simplify]}\\ \text{ = }{\sec }^{2}x+{\tan }^{2}x\text{ [multiply]}\end{array}$

Hence, the simplest form of the expression ${\sec }^{4}x-{\tan }^{4}x$ is option (f) ${\sec }^{2}x+{\tan }^{2}x$ .