SOLUTION: Show all steps necessary to verify the trigonometric identity: 1+tan^2x -------- = csc^2x tan^2x

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Question 816769: Show all steps necessary to verify the trigonometric identity:
1+tan^2x
-------- = csc^2x
tan^2x
Answer by jsmallt9(3758) About Me  (Show Source):
You can put this solution on YOUR website!
First of all, please do not try to use fraction bars when you post. Most of the time they look so bad they are hard to understand. Instead, put parentheses around the numerator and around the denominator and separate them with a slash:
(1+tan^2(x))/tan^2(x) = csc^2(x)

%281%2Btan%5E2%28x%29%29%2Ftan%5E2%28x%29+=+csc%5E2%28x%29
When verifying identities and you're not sure what to do, try changing sec's, csc's, tan's and cot's into expressions of sin and/or cos. This often makes the path clearer. Changing the tan's to sin/cos:

Now we can multiply the numerator and denominator by cos%5E2%28x%29 (to eliminate the fractions within the larger fraction:

Using the Distributive Property on top we get:
%28cos%5E2%28x%29%2Bsin%5E2%28x%29%29%2Fsin%5E2%28x%29+=+csc%5E2%28x%29
We should recognize that the numerator is 1:
1%2Fsin%5E2%28x%29+=+csc%5E2%28x%29
And since sin and csc are reciprocals of each other, the left side is equal to the right:
csc%5E2%28x%29+=+csc%5E2%28x%29