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_________

Options

A. 0
B. 1
C. 2
D. 3

Answer

C

Explanation

The given equation is
`"cos"^(2)(x + (pi)/(6)) + "cos"^(2) x - 2 "cos" (x + (pi)/(6)) "cos" (pi)/(6) = "sin"^(2) (pi)/(6)`
`rArr "cos"^(2)(x + (pi)/(6)) + "cos"^(2) x - "sin"^(2)(pi)/(6) - 2"cos"(x + (pi)/(6))"cos"(pi)/(6) = 0`
`rArr "cos"^(2) (x + (pi)/(6)) + "cos" (x + (pi)/(6))"cos" (x - (pi)/(6))-2"cos" (x + (pi)/(6)) "cos"(pi)/(6) =0`
`"cos"(x + (pi)/(6)){"cos"(x + (pi)/(6)) + "cos"(x - (pi)/(6)) - 2"cos"(pi)/(6)} = 0`
`rArr "cos" (x + (pi)/(6)) {2"cos" x"cos"(pi)/(6)-2"cos"(pi)/(6)} = 0`
`rArr 2" cos"(x + (pi)/(6)) "cos"(pi)/(6)("cos" x -1) =0`
`rArr "cos" (x + (pi)/(6))("cos" x -1) =0`
`rArr x +(pi)/(6) = +-(pi)/(2) " or", x = 0`
`rArr x= (pi)/(3), -(2pi)/(3), 0`
`rArr x =0, (pi)/(3) " " [because x in (-pi//2, pi//2]`