Given equation is
cos 4x = 1 + k sin2x cos2x
Now consider the LHS of the equation,
cos 4x = 2cos2 2x – 1
[Formula for Cos 2x = 2cos2 x – 1]
= 2[2cos2 x - 1]2 – 1
= 2[(2cos2 x)2 – 2 × (2 cos2 x) × (1) + (1)2] – 1
[Applying (a-b)2 = a2 – 2ab + b2 formula]
= 2[4cos4 x – 4cos2 x +1] -1
= 8 cos4 x – 8cos2 x +2 – 1
= 8cos2 x (cos2 x – 1) + 1
= 8cos2 x (-sin2 x) +1
= - 8cos2 x sin2 x + 1
Now as per the LHS cos 4x = - 8cos2 x sin2 x + 1 -------- (1)
Comparing LHS with the RHS,
cos 4x = 1 - 8cos2 x sin2 x = 1 + k sin2x cos2x
by comparing we get k = -8