Calculus (3rd Edition)

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

Chapter 3 - Differentiation - 3.6 Trigonometric Functions - Exercises - Page 140: 30


$$ y=(2-\sqrt{2}) x+\left[\frac{\sqrt{2} \pi}{4}+\sqrt{2}-1-\frac{\pi}{2}\right]$$

Work Step by Step

Given $$y=\csc x-\cot x, \quad x=\frac{\pi}{4}$$ Since $y(\pi/4)= \sqrt{2}-1$ and \begin{align*} y'&=-\csc x\cot x+\csc^2 x\\ y'\bigg|_{x=\pi/4}&= (2-\sqrt{2}) \end{align*} Then the tangent line equation is given by \begin{align*} \frac{y-y_1}{x-x_1}&=m\\ \frac{y- ( \sqrt{2}-1)}{x-\pi/4}&=(2-\sqrt{2})\\ y-(\sqrt{2}-1)&=(2-\sqrt{2})\left(x-\frac{\pi}{4}\right) \end{align*} Hence $$ y=(2-\sqrt{2}) x+\left[\frac{\sqrt{2} \pi}{4}+\sqrt{2}-1-\frac{\pi}{2}\right]$$
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