Prove that: 8^ - 8^ 8^ + 8^ = 37^ - Vidyadip

Question

Prove that:$\frac{\cos8^\circ-\sin8^\circ}{\cos8^\circ+\sin8^\circ}=\tan37^\circ$$ $

✓

Answer

$\text{L.H.S}=\frac{\cos8^\circ-\sin8^\circ}{\cos8^\circ+\sin8^\circ}$dividing num erator and demomintor by $\cos9^\circ$we get,
$\frac{\frac{\cos8^\circ}{\cos8^\circ}-\frac{\sin8^\circ}{\cos8^\circ}}{\frac{\cos8^\circ}{\cos8^\circ}+\frac{\sin8^\circ}{\cos8^\circ}}{}$$\Big[\because\tan\theta=\frac{\sin\theta}{\cos\theta}\Big]$ $=\frac{1-\tan8^\circ}{1+\tan8^\circ}$ $\big[\tan45^\circ=1\big]$ $=\frac{\tan8^\circ-\tan8^\circ}{1+\tan8^\circ\times\tan8^\circ}$ $\Big[\because\tan\text{(A-B)=}\frac{\tan\text{A}-\tan\text{B}}{1+\tan\text{A}\tan\text{B}}\Big]$ $=\tan(45^\circ+8^\circ)$ $=\tan37^\circ$ $=\text{R.H.S}$ $\therefore\text{L.H.S}=\text{R.H.S}$ Hence proved.

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