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Q.
If $a$ is a fixed non-zero constant, then the derivative of
$\frac{\sin (x+a)}{\cos x}$ is
Limits and Derivatives
Solution:
Let $y=\frac{\sin (x+ a)}{\cos x}=\frac{\sin x \cos a+\cos x \sin a}{\cos x}$
$[\because \sin (A+B)=\sin A \cos B+\cos A \sin B]$
$=\frac{\sin x \cos a}{\cos x}+\frac{\cos x \sin a}{\cos x} =\cos a \tan x+\sin a$
Differentiating y w.r.t. x, we get
$\frac{d y}{d x}=\cos a \frac{d}{d x}(\tan x)+\frac{d}{d x}(\sin a)$
$=\cos a \sec ^{2} x+0=\frac{\cos a}{\cos ^{2} x}$