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If $\cos \alpha + \cos \beta = 0 = \sin \alpha + \sin \beta$ , then prove that $\cos 2 \alpha + \cos 2 \beta = -2\cos ( \alpha + \beta )$ . [Hint: $(\cos \alpha + \cos \beta )\textsuperscript{2} - (\sin \alpha + \sin \beta )\textsuperscript{2} = 0$ ]

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Given that: $\cos \alpha +\cos \beta =0~~ and \sin \alpha +\sin \beta =0 \\\\$

$\\ ~So, \left( \cos \alpha +\cos \beta \right) ^{2} - \left( \sin \alpha +\sin \beta \right) ^{2}=0 \\\\ ~~ \left( \\cos ^{2} \alpha +\\cos ^{2} \beta +2 \cos \alpha \cos \beta \right) - \left( \sin^{2} \alpha +\\sin ^{2} \beta +2\sin \alpha \sin \beta \right) =0 \\\\ ~ \left( \\cos ^{2} \alpha - \\sin ^{2} \alpha \right) + \left( \\cos ^{2} \beta - \\sin ^{2} \beta \right) +2 \left( \cos \alpha \cos \beta - \sin \alpha \sin \beta \right) =0 \\\\ \\cos 2 \alpha +cos 2 \beta +2 \cos \left( \alpha + \beta \right) =0 \\\\ \\cos 2 \alpha +\cos2 \beta = - 2 \cos \left( \alpha + \beta \right) \\\\$

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