Prove that: 105^ + 105^ = 45^ - Vidyadip

Question

Prove that: $\sin105^\circ+\cos105^\circ=\cos45^\circ$

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Answer

$\text{LHS}=\sin105^\circ+\cos105^\circ$ $\sin105^\circ+\cos(90^\circ+15^\circ)$ $\sin105^\circ-\sin15^\circ$ $=\ 2\sin\Big(\frac{105^\circ-15^\circ}{2}\Big)\cos\Big(\frac{105^\circ+15^\circ}{2}\Big)$ $=\ 2\sin45^\circ\cos60^\circ$ $=\ 2\frac{1}{\sqrt2}\frac{1}{2}$ $=\ \frac{1}{\sqrt2}$ $=\ \cos45^\circ$

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