SOLUTION: 1/tan^2pie/8 - 1/sin^2pie/8 Evalulate

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Question 1208308: 1/tan^2pie/8 - 1/sin^2pie/8
Evalulate
Found 2 solutions by mccravyedwin, ikleyn:
Answer by mccravyedwin(409) About Me  (Show Source):
You can put this solution on YOUR website!
I just did that. You posted the same problem twice.

Edwin

Answer by ikleyn(52879) About Me  (Show Source):
You can put this solution on YOUR website!
.
1/tan^2pie/8 - 1/sin^2pie/8
Evaluate
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        I will evaluate an expression   1%2Ftan%5E2%28x%29 - 1%2Fsin%5E2%28x%29   for arbitrary  x =/= 0, +/-pi, +/-2pi, . . . 


We have  1%2Ftan%5E2%28x%29} = cos%5E2%28x%29%2Fsin%5E2%28x%29.


THEREFORE,  1%2Ftan%5E2%28x%29 - 1%2Fsin%5E2%28x%29 = cos%5E2%28x%29%2Fsin%5E2%28x%29 - 1%2Fsin%5E2%28x%29 = %28cos%5E2%28x%29-1%29%2Fsin%5E2%28x%29.


The numerator is  -sin%5E2%28x%29, by Pythagoras.  THEREFORE, the last fraction in the previous line is -1.


ANSWER.  The expression  1%2Ftan%5E2%28x%29 - 1%2Fsin%5E2%28x%29  is identically equal to -1 

         for all values of x different from 0, +/-pi, +/-2pi, . . . 

         In particular,  1%2Ftan%5E2%28pi%2F8%29 - 1%2Fsin%5E2%28pi%2F8%29  is -1.

Solved for much more general expression, in simplest possible way, in 3 lines with complete explanations.