Trigonometry (11th Edition) Clone

Published by Pearson
ISBN 10: 978-0-13-421743-7
ISBN 13: 978-0-13421-743-7

Chapter 5 - Trigonometric Identities - Section 5.6 Half-Angle Identities - 5.6 Exercises - Page 243: 45


$$\sec^2\frac{x}{2}=\frac{2}{1+\cos x}$$ The equation is verified to be an identity below.

Work Step by Step

$$\sec^2\frac{x}{2}=\frac{2}{1+\cos x}$$ We start with the left side $$A=\sec^2\frac{x}{2}$$ $$A=\frac{1}{\cos^2\frac{x}{2}}$$ We know that $\cos2x=2\cos^2x-1$, which means $\cos^2x=\frac{\cos 2x+1}{2}$. Therefore, $\cos^2\frac{x}{2}$ can also be written in the same way: $\cos^2\frac{x}{2}=\frac{\cos x+1}{2}$ Therefore, $$A=\frac{1}{\frac{\cos x+1}{2}}$$ $$A=\frac{2}{\cos x+1}$$ The equation has been proved to be an identity.
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