Question

tan square alpha is equal to cos square beta minus sin square beta prove that cos square alpha minus sin square alpha is equal to tansquare beta​

Answer 1

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Answer 2

Answer:

Given: $tan^2\,\alpha=cos^2\,\beta-sin^2\,\beta$

To prove: $cos^2\,\alpha-sin^2\,\alpha=tan^2\beta$

First consider,

$tan^2\,\alpha=cos^2\,\beta-sin^2\,\beta$

$tan^2\,\alpha=cos\,2\beta$   ( using, cos 2x = cos²x - sin²x )

Now,

LHS

$=cos^2\,\alpha-sin^2\,\alpha$

$=cos\,2\alpha$

$=\frac{1-tan^2\,\alpha}{1+tan^2\,\alpha}$

$=\frac{1-cos\,2\beta}{1+cos\,2\beta}$

$=\frac{2sin^2\,\beta}{2cos^2\,\beta}$

$=\frac{sin^2\,\beta}{cos^2\,\beta}$

$=tan^2\,\beta$

$=RHS$

Hence Proved.