Question

How do you verify `(cotx/sec^2x) + (cotx/csc^2x)= cotx`?

Answer 1

Verify trig expression.
`(cot x/sec^2 x) + (cot x/csc^2 x) = cot x`

Explanation:

First term of left side:
`((cos x/(sin x))/(1/(cos^2 x))) = (cos^3 x)/(sin x)` (1)

Second term of left side:
`((cos x)/sin x)/(1/sin^2 x) = (cos x.sin^2 x) /(sin x)` (2).

Add (1) and (2)
`((cos^3 x)/(sin x))+ (cos x sin^2 x)/(sin x) =`
`= (cos x)/(sin x)(cos^2 x + sin^2 x) = (cos x)/(sin x) = cot x`

Answer 2

Determination is done by following method.

Explanation:

LHS= `cotx/sec^2x+cotx/csc^2x`

`" "color(white)(a)=cotx(1/sec^2x+1/csc^2x)`

`" "color(white)(a)=cotx(cos^2x+sin^2x)`

`" "color(white)(a)=cotx`

`" "color(white)(a)=RHS`