If `tan 3 theta + tan theta =2 tan 2 theta`, then `theta` is equal to `(n in Z)`

Options

A. `n pi`
B. `(n pi)/4`
C. `(n pi)/2`
D. none of these

Answer

A

Explanation

`tan 3 theta+tan theta =2 tan 2 theta`
or `tan 3 theta - tan 2 theta = tan 2 theta - tan theta`
or `(sin (3 theta - 2 theta))/(cos 3 theta cos 2 theta)=(sin (2 theta - theta))/(cos 2 theta cos theta)`
or `sin theta (2 sin theta sin 2 theta )=0`
or `sin theta =0 or sin 2 theta =0`
`implies theta = npi or 2 theta = npi, n in Z`
But `theta = n pi//2` is rejected because when `n` is even, i.e., 2r then `theta=r pi`.
Then `theta = npi, n in I` is the only solution.