$t a n x + t a n 2 x + t a n 3 x = t a n x . t a n 2 x . t a n 3 x$
find its general solution $n \prod$

x = n \prod

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x = n \prod + \frac{\prod}{3}

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x = 2 n \prod

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x =

\frac{n \prod}{3}

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Solution

The correct option is D x = $\frac{n \prod}{3}$
$T a n x + T a n 2 x + T a n 3 x = T a n x T a n 2 x T a n 3 x$ _ (1) (Given)
$T a n (3 x) = T a n (x + 2 x) = \frac{T a n x + T a n 2 x}{1 - T a n x T a n 2 x}$
$T a n 3 x - T a n x T a n 2 x T a n 3 x = T a n x + T a n 2 x$
$T a n x + T a n 2 x - T a n 3 x = - T a n x T a n 2 x T a n 3 x$ _ (2)
(1) + (2) $\Rightarrow - T a n 3 x = 0 \Rightarrow T a n 3 x = 0$
$\Rightarrow 3 x = K \prod$
$x = \frac{k \prod}{3}$

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