please, prove that (1-2cos^2x)/(sinx*cosx) = tanx-cotx

Question

I'm stumped on this one! help please!

Answer 1

Kaylin,

 

First stay away from those values of x for which the denominator becomes zero and makes the division meaningless.  Therefore, sin x ≠ 0 (or x ≠ 0^o or 180^o) and cos x ≠ 0 (or x ≠ 90^o or 270^o).

 

And recall the relationship for all angles x that  [ sin²x + cos²x] = 1.  So onto your proof:

 

[1 - 2 cos²x]/(sin x)(cos x) = [(1 - cos²x) - cos²x]/(sin x)(cos x) = [sin²x - cos²x]/(sin x)(cos x)

                                        = sin²x/(sin x)(cos x) - cos²x/(sin x)(cos x) = sin x/cos x   -   cos x/sin x

                                        = tan x  -  cot x