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 - Summary Exercises on Verifying Trigonometric Identities - Page 245: 24


$$2\cos^2\frac{x}{2}\tan x=\tan x+\sin x$$ 2 sides are equal as proved below, so the equation is an identity.

Work Step by Step

$$2\cos^2\frac{x}{2}\tan x=\tan x+\sin x$$ Again, we examine from the left side. $$X=2\cos^2\frac{x}{2}\tan x$$ - For $\cos^2\frac{x}{2}$: From the half-angle identity for cosine: $$\cos\frac{x}{2}=\pm\sqrt{\frac{1+\cos x}{2}}$$ Thus, $$\cos^2\frac{x}{2}=\frac{1+\cos x}{2}$$ (After doubling, the sign is always positive) - For $\tan x$: From the quotient identity: $$\tan x=\frac{\sin x}{\cos x}$$ Apply them back to $X$: $$X=2\times\frac{1+\cos x}{2}\times\frac{\sin x}{\cos x}$$ $$X=\frac{(1+\cos x)\sin x}{\cos x}$$ $$X=\frac{\sin x+\sin x\cos x}{\cos x}$$ Now we separate the fraction. $$X=\frac{\sin x}{\cos x}+\frac{\sin x\cos x}{\cos x}$$ $$X=\tan x+\sin x$$ (as $\tan x=\frac{\sin x}{\cos x}$) Therefore, $$2\cos^2\frac{x}{2}\tan x=\tan x+\sin x$$ 2 sides are equal, so the equation is verified to be an identity.
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