# Question #4d51c

Jul 21, 2016

$L H S = \sin x \left(1 + \tan x\right) + \cos x \left(1 + \cot x\right)$

$= \sin x \left(1 + \tan x\right) + \cos x \left(1 + \frac{1}{\tan} x\right)$

$= \sin x \left(1 + \tan x\right) + \cos x \left(\frac{1 + \tan x}{\tan} x\right)$

$= \left(1 + \tan x\right) \left(\sin x + \cos \frac{x}{\tan} x\right)$

$= \left(1 + \tan x\right) \left(\sin x + \cos \frac{x}{\sin \frac{x}{\cos} x}\right)$

$= \left(1 + \tan x\right) \left(\sin x + {\cos}^{2} \frac{x}{\sin} x\right)$

$= \left(1 + \tan x\right) \left(\frac{{\sin}^{2} x + {\cos}^{2} x}{\sin} x\right)$

$= \frac{1 + \tan x}{\sin} x$

$= \frac{1}{\sin} x + \frac{\cancel{\sin x}}{\cos} x \cdot \frac{1}{\cancel{\sin x}}$

$= \csc x + \sec x = R H S$

Proved

Jul 21, 2016

$L H S = \sin x \left(1 + \tan x\right) + \cos x \left(1 + \cot x\right)$

$= \sin x + \frac{\sin x \cdot \sin x}{\cos} x + \cos x + \frac{\cos x \cdot \cos x}{\sin} x$

$= \sin x + {\sin}^{2} \frac{x}{\cos} x + \cos x + {\cos}^{2} \frac{x}{\sin} x$

$= \sin x + \frac{1 - {\cos}^{2} x}{\cos} x + \cos x + \frac{1 - {\sin}^{2} x}{\sin} x$

$= \sin x + \frac{1}{\cos} x - {\cos}^{2} \frac{x}{\cos} x + \cos x + \frac{1}{\sin} x - {\sin}^{2} \frac{x}{\sin} x$

$= \cancel{\sin} x + \sec x - \cancel{\cos} x + \cancel{\cos} x + \csc x - \cancel{\sin} x$

$\sec x + \csc x = R H S$