Question

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

Answer 1

Recall the definitions of tangent and cotangent:

`\tan x=(\sin x)/(\cos x)`

`\cot x=\frac1{\tan x}=(\cos x)/(\sin x)`

Then

`\sin x(\cot x+\tan x)=\sin x\left((\cos x)/(\sin x)+(\sin x)/(\cos x)\right)=\cos x+(\sin^2x)/(\cos x)`

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

`\sin x(\cot x+\tan x)=\cos x+(1-\cos^2x)/(\cos x)=\cos x+\frac1{\cos x}-\cos x=\frac1{\cos x}`

and the definition of secant,

`\sec x=\frac1{\cos x}`