Write the following functions in the simplest form:

    8. \tan^{-1}\left(\frac{\cos x -\sin x }{\cos x + \sin x} \right ),\;\; \frac{-\pi}{4} < x < \frac{3\pi}{4}

Answers (1)

Given \tan^{-1}\left(\frac{\cos x -\sin x }{\cos x + \sin x} \right )   where x\:\epsilon\:( \frac{-\pi}{4} < x < \frac{3\pi}{4})

So,

=\tan^{-1}\left(\frac{\cos x -\sin x }{\cos x + \sin x} \right )  

Taking \cos x common from numerator and denominator.

We get: 

=\tan^{-1}\left(\frac{1 -(\frac{\sin x}{\cos x}) }{1+(\frac{\sin x}{\cos x}) } \right )

=\tan^{-1}\left(\frac{1 - \tan x }{1+\tan x } \right )

\tan^{-1}(1) - \tan^{-1}(\tan x)         as, \left [ \because \tan^{-1}x - \tan^{-1}y = \frac{x - y}{1 + xy} \right ]

\frac{\pi}{4} - x is the simplest form.

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