Prove that: ( + )^2+( - )^2=4 ^2 x+y 2 - Vidyadip

Question

Prove that:
$(\cos\text{x}+\cos\text{y})^2+(\sin\text{x}-\sin\text{y})^2=4\cos^2\frac{\text{x}+\text{y}}{2}$

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Answer

$\text{L.H.S.}=(\cos\text{x}+\cos\text{y})^2+(\sin\text{x}-\sin\text{y})^2$$=\cos^2\text{x}+\cos^2\text{y}+2\cos\text{x}\cos\text{y}+\sin^2\text{x}+\sin^2\text{y}-2\sin\text{x}\sin\text{y}$
$=(\cos^2\text{x}+\sin^2\text{x})+(\cos^2\text{y}+\sin^2\text{y})+2(\cos\text{x}\cos\text{y}-\sin\text{x}\sin\text{y})$
$=1+1+2\cos(\text{x}+\text{y})\ [\cos(\text{A}+\text{B})=(\cos\text{A}\cos\text{B}-\sin\text{A}\sin\text{B})]$
$=2+2\cos(\text{x}+\text{y})$
$=2[1+\cos(\text{x}+\text{y})]$
$=2\Big[1+2\cos^2\Big(\frac{\text{x}+\text{y}}{2}\Big)-1\Big]\ [\cos2\text{A}=2\cos^2\text{A}-1]$
$=4\cos^2\frac{\text{x}+\text{y}}{2}=\text{R.H.S.}$

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