Question

If $a$ is a fixed non-zero constant, then the derivative of
$\frac{\sin (x+a)}{\cos x}$ is

• A. $\frac{\cos a}{{\cos }^{2}x}$
• B. $\frac{-\cos a}{{\cos }^{2}x}$
• C. $\frac{\sin a}{{\cos }^{2}x}$
• D. $\frac{-\sin a}{{\cos }^{2}x}$

Answer 1

Answer: (A)

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
$\frac{dy}{dx}=\cos a\frac{d}{dx}(\tan x)+\frac{d}{dx}(\sin a)$
$=\cos a{\sec }^{2}x+0=\frac{\cos a}{{\cos }^{2}x}$