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: 54


$$\cot\frac{A}{2}=\frac{1+\cos A}{\sin A}$$

Work Step by Step

$$\tan\frac{A}{2}=\frac{\sin A}{1+\cos A}$$ - From Reciprocal Identities, we have: $$\cot\theta=\frac{1}{\tan\theta}$$ Therefore, we can apply the identity for angle $\frac{A}{2}$ in place of $\theta$. $$\cot\frac{A}{2}=\frac{1}{\tan\frac{A}{2}}$$ Thus, $$\cot\frac{A}{2}=\frac{1}{\frac{\sin A}{1+\cos A}}$$ $$\cot\frac{A}{2}=\frac{1+\cos A}{\sin A}$$ That is the identity for $\cos\frac{A}{2}$.
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