Question

Prove that Cos 2 alpha into cos 2 beta + sin square alpha minus beta minus sin square alpha + beta is equal to cos 2 into alpha + beta

Answer 1

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Answer 2

Answer:

To Prove: $cos\,2\alpha\;cos\,2\beta+sin^2\,(\alpha-\beta)-sin^2\,(\alpha+\beta)=cos\,2(\alpha+\beta)$

Consider,

$LHS\:=\:cos\,2\alpha\;cos\,2\beta+sin^2\,(\alpha-\beta)-sin^2\,(\alpha+\beta)$

now using,

sin(a+b) = sin a cos b + cos a sin b

sin(a-b) = sin a cos b - cos a sin b

$=cos\,2\alpha\;cos\,2\beta+(sin\,\alpha\:cos\,\beta-cos\,\alpha\:sin\,\beta)^2-(sin\,\alpha\:cos\,\beta+cos\,\alpha\:sin\,\beta)^2$

using, a² - b² = ( a - b )( a + b )

$=cos\,2\alpha\;cos\,2\beta+(sin\,\alpha\:cos\,\beta-cos\,\alpha\:sin\,\beta+sin\,\alpha\:cos\,\beta+cos\,\alpha\:sin\,\beta)(sin\,\alpha\:cos\,\beta-cos\,\alpha\:sin\,\beta-(sin\,\alpha\:cos\,\beta+cos\,\alpha\:sin\,\beta))$

$=cos\,2\alpha\;cos\,2\beta+(2\,sin\,\alpha\:cos\,\beta)(-2cos\,\alpha\:sin\,\beta)$

$=cos\,2\alpha\;cos\,2\beta-4\,sin\,\alpha\:cos\,\beta\:cos\,\alpha\:sin\,\beta$

$=cos\,2\alpha\;cos\,2\beta-(2\,sin\,\alpha\:cos\,\alpha)(2\:cos\,\beta\:sin\,\beta)$

$=cos\,2\alpha\;cos\,2\beta-sin\,2\alpha\:sin\,2\beta$

$=cos\,(2\alpha+2\beta)$

$=cos\,2(\alpha+\beta)$

$=RHS$