Prove that: 36^ + 9^ + 36^ + 9^ =1 - Vidyadip

Question

Prove that: $\tan36^\circ+\tan9^\circ+\tan36^\circ+\tan9^\circ=1$

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Answer

We have, $45^\circ=9^\circ+36^\circ$ $\Rightarrow\tan45^\circ=\tan(9^\circ+36^\circ)$ $\Rightarrow1=\frac{\tan9^\circ+\tan36^\circ}{1-\tan9^\circ\tan36^\circ}$ $\Big[\because\tan\text{(A}+\text{B)}=\frac{\tan\text{A}+\tan\text{B}}{1-\tan\text{A}\tan\text{B}}\Big]$ $\Rightarrow 1-\tan9^\circ\tan36^\circ=\tan9^\circ+\tan36^\circ$ $\Rightarrow 1=\tan9^\circ+\tan36^\circ+\tan9^\circ\tan36^\circ$ $\Rightarrow \tan9^\circ+\tan36^\circ+\tan9^\circ\tan36^\circ=1$ Hence proved.

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