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Prove that (cos x + cos y)^(2) + (sin ...

Prove that `(cos x + cos y)^(2) + (sin x + sin y)^(2) = 4 cos^(2) ((x-y)/(2))`

Text Solution

Verified by Experts

We have
`LHS = (cos x + cos y)^(2) + (sin x + sin y)^(2)`
`={2 cos ((x+y)/(2)) cos ((x-y)/(2))^(2)} + {2sin ((x+y)/(2)) cos ((x-y)/(2))}^(2)`
` = 4 cos^(2) ((x+y)/(2)) cos^(2) ((x-y)/(2)) + 4 sin^(2) ((x+y)/(2)) cos^(2) ((x-y)/(2))`
`=4 cos^(2) ((x-y)/(2)).{cos^(2) ((x+y)/(2)) + sin^(2) ((x+y)/(2))}`
` =4 cos^(2) ((x-y)/(2)) = RHS`
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