Question

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Answer 1

$\blue{\bold{\underline{\underline{Be Brainly }}}}$

Answer 2

$\huge\underline\mathfrak\color{olive}Given$

$\sf{\cos\theta + \sin\theta = \sqrt{2}\sin\theta }$

$\huge\underline\mathfrak\color{olive}To \: prove$

$\sf{\cos\theta + \sin\theta = \sqrt{2} \cos\theta}$

$\huge\underline\mathfrak\color{olive} Solution$

$\sf{\cos\theta + \sin\theta = \sqrt{2} \sin\theta}$

$\sf{Now, \: by \: squaring \: both \: sides \: we \: get .}$

$\sf{\implies \cos^{2} + \sin ^{2} - 2\cos\theta\sin\theta = 2 {sin}^{2}\theta }$

$\sf{\implies {cos}^{2} = 2 {sin}^{2} \theta - {sin}^{2} + 2\cos\theta\sin\theta}(By \: transposition)$

$\sf{\implies {cos}^{2}\theta = {sin}^{2} \theta + 2\cos\theta\sin\theta }$

$\sf{By \: adding \: {cos}^{2} \theta \: to \: the \: both \: sides }$

$\sf{\implies 2 {cos}^{2} \theta = {cos}^{2} \theta + {sin}^{2}\theta + 2\cos\theta + 2\cos\theta\sin\theta}$

$\sf{\implies 2 {cos}^{2} \theta = ( {cos\theta + \sin\theta})^{2} }$

$\sf{\implies \sqrt{2}\cos\theta = \cos\theta + \sin\theta }$

Hence, proved.

$\huge\underline\mathfrak\color{olive}ThankYou$