Let S(θ) denote the sum of coefficients in the expansion of 2−xsin⁡ θ+x2cos⁡ θ2n.Maximum value of S(θ) is

S(θ)=(2−sin⁡ θ+cos⁡ θ)2nMaximum possible value of2−sin⁡ θ+cos⁡ θ=2−2cos ⁡ π4sin⁡ θ−sin⁡ π4cos⁡ θ=2−2sin⁡ (θ−π/4) is 2+2=22 ∴ Maximum possible value of S(θ) is 23/22n=8n