SOLUTION: What is the exact value of cos(1/2*tan^-1(-5))?

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Question 387971: What is the exact value of cos(1/2*tan^-1(-5))?
Answer by robertb(5830) About Me  (Show Source):
You can put this solution on YOUR website!
Let alpha+=+tan%5E%28-1%29%28-5%29%2F2. We have to find cos%28alpha%29. Then 2alpha+=+tan%5E%28-1%29%28-5%29, or tan%282alpha%29+=+-5. From the identity %28tan%282alpha%29%29%5E2+%2B+1+=+%28sec%282alpha%29%29%5E2, we get 26+=+%28sec%282alpha%29%29%5E2, or sec%282alpha%29+=+sqrt%2826%29.
Secant of alpha is positive because tan%282alpha%29+=+-5 means 2alpha is in the 2nd quadrant.
Hence cos%282alpha%29+=+1%2Fsqrt%2826%29, or
1%2Fsqrt%2826%29+=+2%28cos%28alpha%29%29%5E2+-+1. (Again using another identity, the double -angle identity for cosine.)
%28cos%28alpha%29%29%5E2+=+%28sqrt%2826%29+%2B+1%29%2F%282sqrt%2826%29%29,
or . Cosine of alpha is positive because 2alpha is in quadrant 2, so when it is divided by 2, alpha must be in quadrant 1.