22.8. Докажите тождество: 5) $$\sin^3 x (1 + ctgx) + \cos^3 x(1 + tgx) = \sin x + \cos x$$

5) $$\sin^3 x (1 + ctgx) + \cos^3 x(1 + tgx) = \sin^3 x (1 + \frac{\cos x}{\sin x}) + \cos^3 x(1 + \frac{\sin x}{\cos x}) = \sin^3 x + \sin^2 x \cos x + \cos^3 x + \cos^2 x \sin x = \sin^2 x(\sin x + \cos x) + \cos^2 x(\cos x + \sin x) = (\sin^2 x + \cos^2 x)(\sin x + \cos x) = 1 \cdot (\sin x + \cos x) = \sin x + \cos x$$. Тождество доказано.