Question

Prove that:
(2-sin(2x))(sin(x) + cos(x)) = 2(sin^3(x) + cos^3(x))

Answer 1

`\text{We use formulas: }\\ \\ 1) ~~ (a + b)(a^2 -ab + b^2) =a^3 + b^3 \\ \\ 2)~~ \sin(2x) = 2\sin x \cos x \\ \\ 3)~~ 1 =\sin^2(x) + cos^2(x) \\ \\ \text{We solve:} \\ \\ \Big(2-\sin(2x)\Big)\Big(\sin(x) + \cos(x)\Big) = 2\Big(\sin^3(x) + cos^3(x)\Big) \\ \\ \Big(2-2\sin(x)\cos(x)\Big)\Big(\sin(x) + \cos(x)\Big) = 2\Big(\sin^3(x) + cos^3(x)\Big) \\ \\ 2\Big(1-\sin(x)\cos(x)\Big)\Big(\sin(x) + \cos(x)\Big) = 2\Big(\sin^3(x) + cos^3(x)\Big)`



`2\Big(\sin^2(x)+\cos^2(x)-\sin(x)\cos(x)\Big)\Big(\sin(x) + \cos(x)\Big) = \\ 2\Big(\sin^3(x) + cos^3(x)\Big) \\ \\ 2\Big(\sin^2(x)-\sin(x)\cos(x)+\cos^2(x)\Big)\Big(\sin(x) + \cos(x)\Big) = \\ 2\Big(\sin^3(x) + cos^3(x)\Big) \\ \\ \boxed{2\Big(\sin^3(x) + cos^3(x)\Big) = 2\Big(\sin^3(x) + cos^3(x)\Big) }`