Prove that: 9^ + 9^ 9^ - 9^ = 54^ - Vidyadip

Question

Prove that: $\frac{\cos9^\circ+\sin9^\circ}{\cos9^\circ-\sin9^\circ}=\tan54^\circ$$ $

✓

Answer

$\text{L.H.S}=$ $\frac{\cos11^\circ+\sin11^\circ}{\cos11^\circ-\sin11^\circ}$ dividing num erator and demomintor by $\cos9^\circ$we get,
$\frac{\frac{\cos9^\circ}{\cos9^\circ}+\frac{\sin9^\circ}{\cos9^\circ}}{\frac{\cos9^\circ}{\cos9^\circ}-\frac{\sin9^\circ}{\cos9^\circ}}{}$ $\Big[\because\tan\theta=\frac{\sin\theta}{\cos\theta}\Big]$ $=\frac{1+\tan9^\circ}{1-\tan9^\circ}$ $\big[\tan45^\circ=1\big]$ $=\frac{\tan45^\circ+\tan9^\circ}{1-\tan45^\circ\times\tan9^\circ}$ $\Big[\because\tan\text{(A+B)=}\frac{\tan\text{A}+\tan\text{B}}{1-\tan\text{A}\tan\text{B}}\Big]$ $=\tan(45^\circ+9^\circ)$ $=\tan54^\circ$ $=\text{R.H.S}$ $\therefore\text{L.H.S}=\text{R.H.S}$ Hence proved.

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