The number of solutions of the equation `cos^2(x+pi/6)+cos^2x-2cos(x+pi/6)dotcospi/6=sin^2pi/6` in interval `((-pi)/2,pi/2)` is_________

Answer

2

Explanation

We have, `(cos (x+pi/6)-cos (pi/6))^(2)=sin^(2)x`
`= 4 sin^(2) (x/2) sin^(2) (x/2+pi/6)=4 sin^(2) (x/2). cos^(2)(x/2)`
`:. "sin"x/2=0` or `sin^(2) (x/2+pi/6)="cos"^(2) x/2`
`rArr x=2mpi, m in Z`
or `pi/2-(x/2+pi/6)=npi pm x/2, n in I`
`:. x=0` or `x=pi/3`