Question

Find the derivative of $\frac{\sin (x+a)}{\cos x}$ where $a$ is a fixed non-zero constant.

Answer 1

Let $f\left(x\right)=\frac{\sin (x+a)}{\cos x}$
Differentiating with respect to $x$
$\Rightarrow f^{\prime }\left(x\right)=\frac{d}{dx}\left(\frac{\sin (x+a)}{\cos x}\right)$
$\Rightarrow f^{\prime }\left(x\right)=\frac{\left[\right(\cos x\left)\frac{d}{dx}\right(\sin (x+a))-\sin (x+a\left)\frac{d}{dx}\cos x\right]}{(\cos x{)}^2}$
$\Rightarrow f^{\prime }\left(x\right)=\frac{\left(\cos x\right)\left(\cos \right(x+a\left)\right)-\left(\sin \right(x+a\left)\right)(-\sin x)}{(\cos x{)}^2}$
$\Rightarrow f^{\prime }\left(x\right)=\frac{\cos (x+a)\cos x+\sin (x+a)\sin x}{(\cos x{)}^2}$
$\Rightarrow f^{\prime }\left(x\right)=\frac{\cos \left[\right(x+a)-x]}{(\cos x{)}^2}$
$[\because \cos (A-B)=\cos A\cos B+\sin A\sin B]$
$\Rightarrow f^{\prime }\left(x\right)=\frac{\cos (x+a-x)}{{\cos }^{2}x}$
$\therefore f^{\prime }\left(x\right)=\frac{\cos a}{{\cos }^{2}x}$