Question

Sin x(cot x +tan x)= sec x

Answer 1

Recall the definitions of tangent and cotangent:

$\tan x=\dfrac{\sin x}{\cos x}$

$\cot x=\dfrac1{\tan x}=\dfrac{\cos x}{\sin x}$

Then

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

Recall that $\sin^2x+\cos^2x=1$:

$\sin x(\cot x+\tan x)=\cos x+\dfrac{1-\cos^2x}{\cos x}=\cos x+\dfrac1{\cos x}-\cos x=\dfrac1{\cos x}$

and the definition of secant,

$\sec x=\dfrac1{\cos x}$