Prove that: 65^ + 65^ = 2 20 ^ - Vidyadip

Question

Prove that: $\sin65^\circ+\cos65^\circ=\sqrt2\cos20^\circ$

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Answer

We have, $\text{LHS}=\sin 65^\circ+\cos65^\circ$ $=\ \sin(45^\circ+20^\circ)+\cos(90^\circ-25^\circ)$ $=\ \sin(45^\circ+20^\circ)+\sin25^\circ$ $=\ \sin(45^\circ+20^\circ)+\sin(45^\circ-20^\circ)$ $=\ 2\sin45^\circ\cos20^\circ$ $=\ 2\times\frac{1}{\sqrt2}\cos20^\circ$ $=\ \frac{\sqrt2\times\sqrt2}{\sqrt2}\times\cos20^\circ$ $=\ \sqrt2\cos20^\circ$ $=\ \text{RHS}$ $\therefore\ \sin60^\circ+\cos65^\circ=\sqrt2\cos20^\circ\ \text{Hence proved}.$

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