Question

Aratati ca pentru orice x∈R:
a) (sin x+cos x)²=1+2 sin x . cos x
b) 1-2 sin²x=2 cos²x-1
c) sin³x+ cos³x=(sin x+cos x)(1-sin x . cos x)

Answer 1

$a) \\ \\ \begin{array}{rcl}(\sin x+\cos x)^2&=&\sin^2 x + 2\cos x \sin x + \cos^2 x \\ &=& \sin^2 x + \cos^2 x + 2\cos x\sin x \\ &=& 1 + 2\cos x \sin x\end{array}\\ \\ b) \\ \\ \begin{array}{rcl}1-2\sin^2x &=& \sin^2 x + \cos^2 x - 2\sin^2 x \\ &=& \sin^2 x - 2\sin^2 x + \cos^2 x \\ &=& -\sin^2 x + \cos^2 x \\ &=& -\sin^2 x - \cos^2 x + \cos^2 x + \cos^2 x \\ &=& -(\sin^2x + \cos^2 x) + 2\cos^2 x \\ &=& -1+2\cos^2 x \\ &=& 2\cos^2 x - 1 \end{array}$

$c) \\ \\ \boxed{a^3 + b^3 = (a+b)(a^2-ab+b^2)} \\ \\ \begin{array}{rcl} \sin^3 x + \cos^3 x &=& (\sin x + \cos x)(\sin^2 x - \sin x \cos x + \cos^2 x) \\ &=& (\sin x + \cos x)(\sin^2x+\cos^2 x - \sin x \cos x) \\ &=& (\sin x + \cos x)(1-\sin x \cos x) \end{array}$