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  4. If cos-1((1-x2/1+x2))+cos-1((1-y2/1+y2))=(π/2), where xy <1, then then

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Q. If $cos^{-1}\left(\frac{1-x^{2}}{1+x^{2}}\right)+cos^{-1}\left(\frac{1-y^{2}}{1+y^{2}}\right)=\frac{\pi}{2}, where\quad xy <1, then$ then

KEAMKEAM 2013Inverse Trigonometric Functions

Solution:

Given $\cos ^{-1}\left(\frac{1-x^{2}}{1+x^{2}}\right)+\cos ^{-1}\left(\frac{1-y^{2}}{1+y^{2}}\right)=\frac{\pi}{2}$
$\Rightarrow \, 2 \tan ^{-1} \,x+2 \tan ^{-1} \,y=\frac{\pi}{2}$
$\Rightarrow \,\tan ^{-1}\left(\frac{x+y}{1-x y}\right)=\frac{\pi}{4}$
$\Rightarrow \, \frac{x+y}{1-x y}=\tan \frac{\pi}{4}=1$
$\Rightarrow \,x+y=1-x y $
$\Rightarrow \,x+y+x y=1$