Prove that: 7 12 + 12 = 5 12 - 12 - Vidyadip

Question

Prove that: $\cos\frac{7\pi}{12}+\cos\frac{\pi}{12}=\sin\frac{5\pi}{12}-\sin\frac{\pi}{12}$

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Answer

$\text{L.H.S}=$ $\cos105^\circ+\cos15^\circ$
$=\cos(90^\circ+15^\circ)+\cos(90^\circ-75^\circ)$
$=-\sin15^\circ+\sin75^\circ$$\Big[\because\cos\big(90+\theta\big)=-\sin\theta\Big]$
$=-\sin75^\circ-\sin15^\circ$$\Big[\cos\big(90-\theta\big)=\sin\theta\Big]$
$=\cos105^\circ+15^\circ=\sin75^\circ-\sin15^\circ$
$=\text{R.H.S}$
$\text{L.H.S}=\text{R.H.S}$
Hence proved.

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