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: 52


$$\cot \left(\frac{\pi}{2}-x\right) =\tan (x) $$

Work Step by Step

We use the know identities to obtain: \begin{aligned} \cot \left(\frac{\pi}{2}-x\right)&=\frac{\cos \left(\frac{\pi}{2}-x\right)}{\sin \left(\frac{\pi}{2}-x\right)}\\ &=\frac{\cos \left(\frac{\pi}{2}\right) \cos (x)+\sin \left(\frac{\pi}{2}\right) \sin (x)}{\sin \left(\frac{\pi}{2}\right) \cos (x)-\cos \left(\frac{\pi}{2}\right) \sin (x)}\\ &=\frac{0 \times \cos (x)+1 \times \sin (x)}{1 \times \cos (x)-0 \times \sin (x)}\\ &=\frac{0+\sin (x)}{\cos (x)-0}\\ &=\frac{\sin (x)}{\cos (x)}=\tan (x) \end{aligned}
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