Number of solutions (s) of the equation `(sin x)/(cos 3x) +(sin 3x)/(cos 9x)+(sin 9x)/(cos 27 x)=0` in the interval `(0, pi/4)` is __________.

Answer

6

Explanation

`(sin x)/(cos 3x)+(sin 3x)/(cos 9x)+(sin 9x)/(cos 27 x)=0`
or `(2 sin x cos x)/(2 cos 3x cos x)+(2 sin 3x cos 3x)/(2 cos 9x cos 3x)+(2 sin 9x cos 9x)/(2 cos 27 x cos 9x)=0`
or `(sin (3x-x))/(2 cos 3x cos x)+(sin (9x-3x))/(2 cos 9x cos 3x)+(sin (27x -9x))/(2 cos 27 x cos 9x)=0`
or `(tan 3x-tan x)+(tan 9x-tan 3x)+(tan 27x-tan 9x)=0`
or `tan 27 x-tan x=0`
or `tan x=tan 27 x`
`rArr 27x=n pi +x, n in I`
or `x=(npi)/26, n in I`.
or `x=pi/26, (2pi)/26, (3pi)/26, (4pi)/26, (5pi)/26, (6pi)/26`
Hence, there are six solutions.