Prove the following : 2 ( 4 -A )= A+ A - Vidyadip

Question

Prove the following : $\sqrt{2} \cos \left(\frac{\pi}{4}-A\right)=\cos A+\sin A$

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Answer

$\begin{aligned}
\text { L.H.S. } & =\sqrt{2} \cos \left(\frac{\pi}{4}-\mathrm{A}\right) \\
& =\sqrt{2}\left(\cos \frac{\pi}{4} \cos \mathrm{A}+\sin \frac{\pi}{4} \sin \mathrm{A}\right) \\
& =\sqrt{2}\left(\frac{1}{\sqrt{2}} \cos \mathrm{A}+\frac{1}{\sqrt{2}} \sin \mathrm{A}\right) \\
& =\frac{\sqrt{2}}{\sqrt{2}}(\cos \mathrm{A}+\sin \mathrm{A}) \\
& =\cos \mathrm{A}+\sin \mathrm{A} \\
& =\text { R.H.S. }
\end{aligned}$

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