prove that 1 - Cos^2x = Cos^2x Tan^2x

Question

prove that 

1 - Cos^2x = Cos^2x Tan^2x

Answer 1

      1 - cos²x = (cos²x)(tan²x)

Recall the following identities:

     sin²x + cos²x = 1     ==>     sin²x = 1 - cos²x

     tan x = sin x/cos x     ==>     tan²x = sin²x/cos²x

Thus,

     1 - cos²x = (cos²x)(tan²x)

        sin²x   = (cos²x)(sin²x/cos²x)

        sin²x   = (cos²x·sin²x)/cos²x

        sin²x   = (cos²x/cos²x)·sin²x

        sin²x   = (1)sin²x

        sin²x   =   sin²x

Answer 2

Taking the identity 1=sin^2+cos^2, we can rewrite the left side of the equation as sin^2+cos^2-cos^2= sin^2.  Thus, we now have sin^2=cos^2*tan^2.  Divide by cos^2 and we have sin^2/cos^2= tan^2.  Thus, tan^2=tan^2 and we're done.  If you'd like, I can elaborate on why the identity 1=sin^2+cos^2 is true or even give a formal proof if you'd like.