SOLUTION: Prove the identity: {{{(Csc^2x-Cot^2x)/(Sec^2x)=Cos^2x}}}

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Question 148633: Prove the identity:
%28Csc%5E2x-Cot%5E2x%29%2F%28Sec%5E2x%29=Cos%5E2x
Answer by Edwin McCravy(20056) About Me  (Show Source):
You can put this solution on YOUR website!
Prove the identity:
%28Csc%5E2x-Cot%5E2x%29%2F%28Sec%5E2x%29=Cos%5E2x

Make two fractions out of the left side:

%28Csc%5E2x%29%2F%28Sec%5E2x%29-%28Cot%5E2x%29%2F%28Sec%5E2x%29

Substitute 1%2F%28Sin%5E2x%29 for Csc%5E2x.
Substitute 1%2F%28Cos%5E2x%29 for both Sec%5E2x's.
Substitute %28Cos%5E2x%29%2F%28Sin%5E2x%29 for Cot%5E2x.



Invert and multiply:



Simplify:

%28Cos%5E2x%29%2F%28Sin%5E2x%29-%28Cos%5E4x%29%2F%28Sin%5E2x%29

Combine the numerators over the common denominator:

%28Cos%5E2x-Cos%5E4x%29%2F%28Sin%5E2x%29

Factor Cos%5E2x out of the numerator:

%28Cos%5E2x%281-Cos%5E2x%29%29%2F%28Sin%5E2x%29

Replace 1-Cos%5E2x by Sin%5E2x

%28Cos%5E2x%28Sin%5E2x%29%29%2F%28Sin%5E2x%29

Cancel the Sin%5E2x's

%28Cos%5E2x%28cross%28Sin%5E2x%29%29%29%2F%28cross%28Sin%5E2x%29%29

Cos%5E2x

Edwin