Verified answer

y=(1+cos²x)^6

dy/dx=6(1+cos²x)^5 * -2sin x cos x

= -12 sin x cos x (1+cos²x)^5

dy/dx = 6 ( 1 + cos ² x )^5 ( 2 cos x ) (- sin x )

dy/dx = - 6 ( 1 + cos ² x )^5 ( sin 2x )