Question

If acos^3θ + 3asin^2θcosθ=m and asin^3θ + 3asinθcos^2θ=n,prove that (m+n)^2/3 + (m-n)^2/3=2a^3

Answer 1

$\bf\implies\:m\:+\:n\:=\:a\: \cos^3\theta\:+\:3\:a\:\cos\theta\:\sin^2\theta\:+\:3\:a\:\cos^2\theta\: \sin\theta\:=\:a\:(\:\sin\theta\:+\:\cos\theta\:)^3$

$\sf\implies\:(\:m\:+\:n)^2/3\:+(\:m-\:n)^2/3\:=\:2\:a^3$