If Δ 1 = x s i n θ c o s θ - s i n θ - x 1 c o s θ 1 x and Δ 2 = x s i n 2 θ c o s 2 θ - s i n 2 θ - x 1 c o s 2 θ 1 x , x ≠ 0 ; then for all θ ∈ 0 ,   π 2 :

Step-by-step solution

Δ 1 = x s i n θ - s i n θ - x c o s θ 1         c o s θ 1 x = x - x 2 - 1 - s i n θ - x s i n θ - c o s θ + c o s θ ( - s i n θ + x c o s θ ) = - x 3 - x + x s i n 2 θ + s i n θ cos ⁡ θ - s i n θ c o s θ + x c o s 2 θ = - x 3 - x + x sin 2 ⁡ θ + cos 2 ⁡ θ = - x 3 Similarly: Δ 2 = - x 3 ∴ Δ 1 + Δ 2 = - 2 x 3