If `(1)/(2) sin^(-1) [(3 sin 2 theta)/(5 + 4 cos 2 theta)] = tan^(-1) x, " then " x =`

Options

A. `tan 3 theta`
B. `3 tan theta`
C. `(1//3) tan theta`
D. `3 cot theta`

Answer

C

Explanation

`(1)/(2) sin^(-1).(3 sin 2 theta)/(5 + 4 cos 2 theta) = tan^(-1) x`
`rArr sin^(-1).((6 tan theta)/(1 + tan^(2) theta))/(5 + 4 ((1-tan^(2) theta)/(1 + tan^(2) theta))) = sin^(-1).(2x)/(1 + x^(2))`
`rArr sin^(-1).(6 tan theta)/(9 + tan^(2) theta) = sin^(-1).(2x)/(1 + x^(2))`
`rArr sin^(-1).(2((tan theta)/(3)))/(1 + ((tan theta)/(3))^(2)) = sin^(-1).(2x)/(1 + x^(2))`
`rArr x = (tan theta)/(3)`