Number of solution(s) satisfying the equation `1/(sinx)-1/(sin2x)=2/(sin4x)` in `[0,4pi]` equals 0 (b) 2 (c) 4 (d) 6

Options

A. 0
B. 2
C. 4
D. 6

Answer

C

Explanation

`1/(sin x)-1/(sin 2x)=2/(sin 4x)`
`rArr (sin 2x- sin x)/(sin x sin 2x)=2/(2 sin 2x cos 2x)`
`rArr 2 sin 2x cos 2x-2 sin x cos 2x=2 sin x`
`rArr sn 4x-sin 3x+sin x=2 sin x`
`rArr sin 4x=sin 3x+sin x`
`rArr 2 sin 2x cos 2x=2 sin 2x cos x`
`rArr 2x=2npi pm x" "("as "sin 2x ne 0)`
`rArr x=(2n pi)/3, n in I" "("as "x=2k pi" is not in domain")`
`n=1, 2, 4, 5, ...`
Thus, four solutions are `(2pi)/3, (4pi)/3, (8pi)/3, (10 pi)/3`.