Calculus (3rd Edition)

Published by W. H. Freeman
ISBN 10: 1464125260
ISBN 13: 978-1-46412-526-3

Chapter 1 - Precalculus Review - 1.4 Trigonometric Functions - Exercises - Page 31: 62


We derive the given identities below.

Work Step by Step

We derive the identities as follows: \begin{align*} \tan (a+b)&=\frac{\sin (a+b)}{\cos (a+b)}\\ &=\frac{\sin a\cos b+\cos a\sin b}{ \cos a\cos b-\sin a\sin b}\\ &= \frac{\frac{\sin a\cos b}{ \cos a\cos b}+\frac{\cos a\sin b}{ \cos a\cos b}}{1-\frac{\sin a\sin b}{ \cos a\cos b}}\\ &= \frac{\tan a+\tan b}{1-\tan a \tan b} \end{align*} and \begin{align*} \cot(a-b)&=\frac{\cos (a-b)}{\sin (a-b)}\\ &=\frac{ \cos a\cos b+\sin a\sin b}{\sin a\cos b-\cos a\sin b}\\ &= \frac{\frac{ \cos a\cos b }{ \sin a\sin b }+\frac{\sin a\sin b}{\sin a\sin b }}{\frac{\sin a\cos b}{ \sin a\sin b}-\frac{\cos a\sin b}{ \sin a\sin b}}\\ &= \frac{\cot a \cot b+1}{\cot b-\cot a} \end{align*}
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