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: 3


$$\tan\frac{x}{2}=\csc x-\cot x$$ As proved in the Work Step by Step, the equation is an identity.

Work Step by Step

$$\tan\frac{x}{2}=\csc x-\cot x$$ The right side would be examined first here. $$X=\csc x-\cot x$$ We can rewrite both $\csc x$ and $\cot x$ according to the following identities: $$\csc x=\frac{1}{\sin x}\hspace{2cm}\cot x=\frac{\cos x}{\sin x}$$ Therefore, $$X=\frac{1}{\sin x}-\frac{\cos x}{\sin x}$$ $$X=\frac{1-\cos x}{\sin x}$$ Now recall 3 half-angle identities for tangent, one of which is $$\tan\frac{x}{2}=\frac{1-\cos x}{\sin x}$$ Therefore, $$X=\tan\frac{x}{2}$$ Hence, $$\tan\frac{x}{2}=\csc x-\cot x$$ The equation is an identity, verified by the fact 2 sides of the equation are proved to be equal.
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