Q. 78 - Precalculus Enhanced with Graphing Utilities

Question

Use properties of the trigonometric functions to find the exact value of each expression. Do not use a calculator.

${\sec }^2{18}^{\mathrm{o}}-{\tan }^2{18}^{\mathrm{o}}$

Step-by-Step Solution

Answer

The exact value of given trigonometry expression is 1.

Step 1. Given Information

Given that the trigonometry expression is ${\sec }^2{18}^{\mathrm{o}}-{\tan }^2{18}^{\mathrm{o}}.$ To find the exact values of given expression using Pythagorean identity.

Step 2. Pythagorean identity

On relating the Pythagorean identity,

$$\begin{gathered} {\tan }^2\mathrm{\theta }+1-{\tan }^2\mathrm{\theta }={\sec }^2\mathrm{\theta }-{\tan }^2\mathrm{\theta } \\ 1={\sec }^2\mathrm{\theta }-{\tan }^2\mathrm{\theta } \\ {\sec }^2{18}^{\mathrm{o}}-{\tan }^{2}18=1 \end{gathered}$$