Question

How do you prove this identity?

Prove:

`cotx=sinxsin(pi/2-x)+cos^2xcotx`

Please and thanks :)

Answer 1

See below.

Explanation:

These are the identities I use in this proof:

`sin(pi/2 - x) = cos x`

`sin^2x + cos^2x =1`

`color(white)x`

`cot x = sin x sin (pi/2 - x) + cos^2 x cot x`

`=sin x cos x + cos^2x cot x`

`=sin x cos x + (cos^2 x * cos x / sin x)`

`=sin x cos x + cos^3x/sin x`

`=(sin^2 x cos x) / sin x + cos^3x/sin x`

`=(sin^2 x cos x + cos ^3x )/sin x`

`=(cos x (sin^2x + cos^2x))/sin x`

`=(cos x *1) /sin x`

`=cos x / sin x`

`=cot x`

`color(white)x cot x = cot x`