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If $2 \tan ^{-1}\left ( \cos \theta \right )=\tan^{-1}\left ( cosec \theta \right ),$ then show that $\theta =\frac{\pi}{4}$ , where n is any integer.

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We have $2 \tan^{-1}\left ( \cos \theta \right )=\tan^{-1}\left ( cosec\, \theta \right )$

$\Rightarrow \tan^{-1}\left ( \frac{2 \cos \theta }{1-\cos^{2}\theta } \right )=\tan^{-1}\left ( 2\, cosec\, \theta \right )$ $\left [ Since \: 2 \tan^{-1}x=\tan^{-1}\left ( \frac{2x}{1-x^{2}} \right ) \right ]$

$\Rightarrow \frac{2 \cos \theta }{\sin^{2}\theta }=2 \, cosec\, \theta$

$\Rightarrow \cot \theta . 2\, cosec\, \theta =2\, cosec\, \theta$

$\Rightarrow \cot \theta =1$

$\Rightarrow \cot \theta =\cot\frac{\pi}{4}$

$\Rightarrow \theta =\frac{\pi}{4}$

Hence Proved

Posted by

infoexpert24

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