Question

How do you verify the identitiy?

`(cot x cos x)/(cot x + cos x) = (cot x - cos x)/(cot x cos x)`

Answer 1

See below.

Explanation:

I always find it helpful rewriting in sine and cosine.

`((cosx/sinx)cosx)/(cosx/sinx + cosx) = (cosx/sinx - cosx)/(cosx/sinx(cosx))`

`(cos^2x/sinx)/((cosx + sinxcosx)/sinx) = ((cosx - cosxsinx)/sinx)/(cos^2x/sinx)`

We get the following after simplifying these two complex fractions

`cos^2x/(cosx + sinxcosx) = (cosx - cosxsinx)/cos^2x`

Simplifying further, using `cos^2x +sin^2x =1`:

`(1- sin^2x)/(cosx(1 + sinx)) = (cosx(1 - sinx))/(1 - sin^2x)`

`((1 + sinx)(1 - sinx))/(cosx(1 + sinx)) = (cosx(1 - sinx))/((1 + sinx)(1 - sinx))`

`(1 - sinx)/cosx = cosx/(1 + sinx)`

If we multiply both sides by cosine, we get:

`(1 - sinx)/cosx * cosx = cosx/(1 + sinx) * cosx`

`1 - sinx = cos^2x/(1 + sinx)`

`1 - sinx = (1 - sin^2x)/(1 + sinx)`

`1 - sinx = ((1 + sinx)(1 - sinx))/(1 + sinx)`

`1- sinx = 1- sinx`

`LHS = RHS`

Hopefully this helps!