We need the solutions of sin 12x + cos 6x = 0.

sin 12x + cos 6x = 0

=> 2*sin 6x*cos 6x + cos 6x = 0

let cos 6x = y

=> 2*sqrt(1 - y^2)*y + y = 0

=> y(2*sqrt(1 - y^2) + 1) = 0

=>...

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We need the solutions of sin 12x + cos 6x = 0.

sin 12x + cos 6x = 0

=> 2*sin 6x*cos 6x + cos 6x = 0

let cos 6x = y

=> 2*sqrt(1 - y^2)*y + y = 0

=> y(2*sqrt(1 - y^2) + 1) = 0

=> y = 0

2*sqrt(1 - y^2) + 1 = 0

=> 4*(1 - y^2) = 1

=> 1 - y^2 = 1/4

=> y^2 = 3/4

=> y = sqrt 3/2

cos 6x = 0 and cos 6x = sqrt 3/2

6x = arc cos 0 and x = (1/6)*arc cos (sqrt 3/2)

x = pi/12 + n*pi/3 and x = pi/36 + n*pi/3

**The solution of the equation is x = pi/12 + n*pi/3 and x = pi/36 + n*pi/6, where n is a positive or negative integer. **