SOLUTION: 1/tan^2pie/8 - 1/sin^2pie/8 Evaluate Thanks

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Question 1208309: 1/tan^2pie/8 - 1/sin^2pie/8
Evaluate
Thanks
Found 3 solutions by mccravyedwin, ikleyn, MathTherapy:
Answer by mccravyedwin(408) About Me  (Show Source):
You can put this solution on YOUR website!

Note: π is spelled "pi", not "pie" like you eat. LOL
  
1%5E%22%22%2Ftan%5E2%28pi%2F8%29-1%5E%22%22%2Fsin%5E2%28pi%2F8%29

Two well known identities are

tan%28expr%281%2F2%29x%29%22%22=%22%22%22%22+%2B-+sqrt%28%281-cos%28x%29%29%2F%281%2Bcos%28x%29%29%29
and
sin%28expr%281%2F2%29x%29%22%22=%22%22%22%22+%2B-+sqrt%28%281-cos%28x%29%29%2F%282%29%29

Squaring both 

tan%5E2%28expr%281%2F2%29x%29%22%22=%22%22%281-cos%28x%29%29%2F%281%2Bcos%28x%29%29
and
sin%5E2%28expr%281%2F2%29x%29%22%22=%22%22%281-cos%28x%29%29%2F2

Substituting pi%2F4

tan%5E2%28pi%2F8%29%22%22=%22%22%281-cos%28pi%2F4%29%29%2F%281%2Bcos%28pi%2F4%29%29
and
sin%5E2%28pi%2F8%29%22%22=%22%22%281-cos%28pi%2F4%29%29%2F2%5E%22%22

Substituting cos%28pi%2F4%29%22%22=%22%22sqrt%282%29%2F2

tan%5E2%28pi%2F8%29%22%22=%22%22%281-sqrt%282%29%2F2%29%2F%281%2Bsqrt%282%29%2F2%29%22%22=%22%22%282-sqrt%282%29%29%2F%282%2Bsqrt%282%29%29
and
sin%5E2%28pi%2F8%29%22%22=%22%22%281-sqrt%282%29%2F2%29%2F2%5E%22%22%22%22=%22%22%282-sqrt%282%29%29%2F4

reciprocating both:

1%5E%22%22%2Ftan%5E2%28pi%2F8%29%22%22=%22%22%282%2Bsqrt%282%29%29%2F%282-sqrt%282%29%29
1%5E%22%22%2Fsin%5E2%28pi%2F8%29%22%22=%22%224%2F%282-sqrt%282%29%29
and
1%5E%22%22%2Ftan%5E2%28pi%2F8%29-1%5E%22%22%2Fsin%5E2%28pi%2F8%29%22%22=%22%22%282%2Bsqrt%282%29%29%2F%282-sqrt%282%29%29-4%2F%282-sqrt%282%29%29%22%22=%22%22%28-2%2Bsqrt%282%29%29%2F%282-sqrt%282%29%29%22%22=%22%22%28-%282-sqrt%282%29%29%29%2F%282-sqrt%282%29%29%22%22=%22%22%22%22=%22%22%28-%28cross%282-sqrt%282%29%29%29%29%2F%28cross%282-sqrt%282%29%29%29%22%22=%22%22-1

Answer = -1

Edwin

Answer by ikleyn(52873) About Me  (Show Source):
You can put this solution on YOUR website!
.
1/tan^2pie/8 - 1/sin^2pie/8
Evaluate
~~~~~~~~~~~~~~~~~


        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.



Answer by MathTherapy(10556) About Me  (Show Source):
You can put this solution on YOUR website!

1/tan^2pie/8 - 1/sin^2pie/8

Evaluate

Thanks 

1/tan^2piepi/8 - 1/sin^2piepi/8
1%2F%28tan%5E%282%29%28pi%2F8%29%29+-+1%2F%28sin%5E%282%29%28pi%2F8%29%29
matrix%281%2C3%2C+1%2F%28tan%5E%282%29%28pi%2F8%29%29%2C+%22=%22%2C+cot%5E2%28pi%2F8%29%29       and      matrix%281%2C3%2C+1%2F%28sin%5E%282%29%28pi%2F8%29%29%2C+%22=%22%2C+csc%5E2%28pi%2F8%29%29 
1%2F%28tan%5E%282%29%28pi%2F8%29%29+-+1%2F%28sin%5E%282%29%28pi%2F8%29%29 then becomes:cot%5E2%28pi%2F8%29+-+csc%5E2%28pi%2F8%29 

IDENTITY: 
              Therefore, matrix%281%2C3%2C+csc%5E2+%28pi%2F8%29%2C+%22=%22%2C+1+%2B+cot%5E2%28pi%2F8%29%29


                             cot%5E2%28pi%2F8%29+-+%281+%2B+cot%5E2%28pi%2F8%29%29 ----- Substituting1+%2B+cot%5E2%28pi%2F8%29 for csc%5E2+%28pi%2F8%29