If f(x)= 2x 1+x^2 , show that f( )= 2 - Vidyadip

Question

If $\text{f(x)}=\frac{2\text{x}}{1+\text{x}^2},$ show that $\text{f}(\tan\theta)=\sin2\theta$

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Answer

We have, $\text{f(x)}=\frac{2\text{x}}{1+\text{x}^2}$ Now, $\text{f}(\tan\theta)=\frac{2(\tan\theta)}{1+\tan^2\theta}$ $=\sin2\theta$ $\Big[\because\ \sin2\theta=\frac{2\tan\theta}{1+\tan^2\theta}\Big]$ $\therefore\text{ f}(\tan\theta)=\sin2\theta$ Hence, proved.

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