Express As Function of x?

sin(θ) = 2x
Then, θ = sin⁻¹(2x)

cos(2θ)
= 1 - 2 sin²(θ)
= 1 - 2 (2x)²
= 1 - 8x²

Hence, θ + cos(2θ) = sin⁻¹(2x) + 1 - 8x²

sin²θ + cos²θ = 1

so, cos²θ = 1 - sin²θ => 1 - (2x)²

i.e. cos²θ = 1 - 4x²

Now, cos2θ = 2cos²θ - 1 => 2(1 - 4x² ) - 1

Hence, cos2θ = 1 - 8x²

so, θ + cos2θ = sin⁻¹(2x) + 1 - 8x²

:)>

sin(theta) = 2x  >>>  theta = invsin(2x)
  2•theta =  2invsin(2x)cos (2•theta)  = 1 - 2 sin^2(theta)theta + cos (2•theta)  =   invsin(2x)  +  cos (2invsin(2x))  <<< answer