Question

Write the function ${\tan }^{-1}\left(\frac{\cos x-\sin x}{\cos x+\sin x}\right),x<\pi$ in the simplest form

Answer 1

$ta{n}^{-1}\left(\frac{cosx-sinx}{cosx+sinx}\right)$

$=ta{n}^{-1}\left(\frac{1-tanx}{1+tanx}\right)$

$=ta{n}^{-1}\left(\frac{tan{45}^0-tanx}{1+tanxtan{45}^0}\right)$

$=ta{n}^{-1}\left(tan\right({45}^0-x\left)\right)$
Now
$tan({45}^0-x)>0$ for
${90}^0>{45}^0-x>0$
Or ${45}^0>x>-{45}^0$.
Since $x<{180}^0$ hence for ${45}^0>x>-{45}^0$
$ta{n}^{-1}\left(tan\right({45}^0-x\left)\right)={45}^0-x$