If cos y = x cos ( a + y ) with cos a = 1 , then d y d x is equal to:

Question

If $\cos y = x \cos (a + y)$ with $\cos a = 1$ , then $\frac{d y}{d x}$ is equal to:

Smart Achievers · Accepted Answer

(2)

Ans. $\frac{\cos^{2} (a + y)}{\sin a}$

Explanation:

Given, $\cos y = x \cos (a + y)$

$\Rightarrow$ $x = \frac{\cos y}{\cos (a + y)}$

$∴$ $\frac{d y}{d x} = \frac{d}{d y} [\frac{\cos y}{\cos (a + y)}]$

$= \frac{\cos (a + y) (- \sin y) - \cos y [- \sin (a + y)]}{\cos^{2} (a + y)}$

$= \frac{\cos y \sin (a + y) - \sin y \cos (a + y)}{\cos^{2} (a + y)}$

$= \frac{\sin ((a + y) - y)}{\cos^{2} (a + y)} = \frac{\sin a}{\cos^{2} (a + y)}$ $[∵ \sin (A - B) = \sin A \cos B - \cos A \sin B]$

Hence, $\frac{d y}{d x} = \frac{1}{\frac{d x}{d y}} = \frac{\cos^{2} (a + y)}{\sin a}$

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