SOLUTION: If tan^2 A = cos^2 B -sin^2 B ,prove that cos^2 A -sin^2 A =tan^2 B

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Question 474966: If tan^2 A = cos^2 B -sin^2 B ,prove that cos^2 A -sin^2 A =tan^2 B
Answer by robertb(5830) About Me  (Show Source):
You can put this solution on YOUR website!
We are given that tan%5E2+%28A%29++=+cos%5E2+%28B%29++-+sin%5E2+%28B%29, or
tan%5E2+%28A%29++=+cos%282B%29.
It is enough to show that tan%5E2+%28B%29++=+cos%282A%29.
Now cos%282B%29+=+2cos%5E2%28B%29+-+1+=+tan%5E2%28A%29
==> 2cos%5E2%28B%29+=+1%2B+tan%5E2%28A%29 ==> 2cos%5E2%28B%29+=+sec%5E2%28A%29
==> cos%5E2%28B%29+=+sec%5E2%28A%29%2F2
==> sec%5E2%28B%29+=+2cos%5E2%28A%29, after taking reciprocals.
==> sec%5E2%28B%29-1+=+2cos%5E2%28A%29-1,
==> tan%5E2%28B%29+=+cos%282A%29, and the statement is proved.