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  4. The value of x satisfying the equation tan-1 x + tan-1((2/3)) tan-1 ((7/4))is equal to

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Q. The value of $x$ satisfying the equation tan$^{-1}$ x + tan$^{-1}$$\left(\frac{2}{3}\right)$ tan$^{-1}$ $\left(\frac{7}{4}\right)$is equal to

KEAMKEAM 2016Inverse Trigonometric Functions

Solution:

We have,
$\tan ^{-1} x+\tan ^{-1}\left(\frac{2}{3}\right) =\tan ^{-1}\left(\frac{7}{4}\right) $
$ \tan ^{-1}\left[\frac{x+\frac{2}{3}}{1-\frac{2}{3}(x)}\right]=\tan ^{-1}\left(\frac{7}{4}\right) \Rightarrow \frac{\frac{3 x+2}{3}}{\frac{3-2 x}{3}}=\frac{7}{4} $
$ \Rightarrow 4(3 x+2)=7(3-2 x) $
$ \Rightarrow 12 x+8=21-14 x $
$ \Rightarrow 26 x=13 \Rightarrow x=\frac{13}{26}=\frac{1}{2} $