Prove that (1+tanx)^2-2tanx=1/(1-sinx)(1+sinx)

[wishiknewit]

Homework Statement

prove that: (1+tanx)^2-2tanx=1/(1-sinx)(1+sinx)

Homework Equations

Main question is does 1/cosx^2 = secx^2? More in my attempt at the solution.

The Attempt at a Solution

(1+tanx)^2-2tanx=1/(1-sinx)(1+sinx)

(1+tanx)(1+tanx)= 1+2tanx+tanx^2
1+2tanx+tanx^2-2tanx=1+tanx^2
1+tanx^2=secx^2

Now to the right side.

1/(1-sinx)(1+sinx)= 1/1-sinx^2 Now i will solve for the denominator.
1-sinx^2=cosx^2 the equation is now:

1/cosx^2. I know that secx=1/cosx my main question is does secx^2=1/cosx^2? This would give me my correct answer but I somehow feel that if you square secx it does not = 1/cosx.

 

[eumyang]

If
$$\frac{1}{\cos x}= \sec x$$
then certainly
$$\frac{1}{\cos^2 x}= \sec^2 x$$
.