The identity we have to prove is: cos x/(sec x - 1) - cos x/ tan^2x = cot^2x

Let's start from the left hand side

cos x/(sec x - 1) - cos x / (tan x)^2

use sec x = 1/cos x and tan x = sin x/ cos x

=> cos x/(1/cos x - 1) - cos x*(cos x)^2/(sin x)^2

=> (cos x)^2/(1 - cos x) - cos x*(cos x)^2/(sin x)^2

=> [(cos x)^2*sin x^2 - (cos x)^3 + (cos x)^4]/(sin x)^2(1 - cos x)

=>[(cos x)^2(1 - (cos x)^2) - (cos x)^3 + (cos x)^4]/(sin x)^2(1 - cos x)

=>[(cos x)^2 - (cos x)^4) - (cos x)^3 + (cos x)^4]/(sin x)^2(1 - cos x)

=>[(cos x)^2 - (cos x)^3](sin x)^2(1 - cos x)

=>(cos x)^2(1 - cos x)/(sin x)^2(1 - cos x)

=> (cos x/ sin x)^2

=> (cot x)^2

which is the right hand side.

**This proves: cos x/(sec x - 1) - cos x/ tan^2x = cot^2x**

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