# The value of  $tan^{-1}\left ( 1 \right )+tan^{-1}\left ( 0 \right )+tan^{-1}\left ( 2 \right )+tan^{-1}\left ( 3 \right )$  is equal to Option 1) $\pi$ Option 2) $\frac{5\pi}{4}$ Option 3) $\frac{\pi}{2}$ Option 4) None of these

As we learnt in

Formulae of Inverse Trigonometric Functions -

$\sin ^{-1}x + \sin ^{-1}y= \sin ^{-1}\left [ x\sqrt{1-y^{2}}+ y \sqrt{1-x^{2}} \right ]$

$\sin ^{-1}x + \sin ^{-1}y=\pi - \sin ^{-1}\left [ x\sqrt{1-y^{2}}+ y \sqrt{1-x^{2}} \right ]$

- wherein

$x\geqslant 0, y\geqslant 0, x^{2}+ y^{2}\leqslant 1$

$x\geqslant 0, y\geqslant 0, x^{2}+y^{2}> 1$

or

Formulae of Inverse Trigonometric Functions -

$\cos ^{-1}x + \cos ^{-1}y = \cos ^{-1}\left \left ( xy- \sqrt{1-x^{2}}\sqrt{1-y^{2}} \right )$

- wherein

$x\geqslant 0, y\geqslant 0$

$\tan ^{-1}(1)+\tan ^{-1}(0)+ \tan ^{-1} (2)+ \tan ^{-1}(3)$

$\Rightarrow \frac{\pi }{4}+0+ \tan ^{-1}\frac{s}{16}$

$\Rightarrow \frac{\pi }{4}+ \tan ^{-1}(-1)=\frac{\pi }{4}+\pi -\frac{\pi }{4}=\pi$

Option 1)

$\pi$

Correct option

Option 2)

$\frac{5\pi}{4}$

Incorrect option

Option 3)

$\frac{\pi}{2}$

Incorrect option

Option 4)

None of these

Incorrect option

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