Question

sa se determine egalitatea (1+tgx)cos^3x+(1+ctgx)sin^3x=sinx+cosx

Answer 1

Membrul stang se poate scrie :

[tex]\left(1+\dfrac{\sin x}{\cos x}\right)\cos^3 x +\left(1+\dfrac{\cos x}{\sin x}\right) \sin^3 x \\\;\\ \\\;\\ = \dfrac{\cos x +\sin x}{\cos x} \cdot \cos^3 x +\dfrac{\sin x +\cos x}{\sin x} \cdot \sin^3 x \\\;\\ =(\sin x+\cos x)\left(\dfrac{\cos^3 x}{\cos x} + \dfrac{\sin^3 x}{\sin x} \right) \\\;\\ = (\sin x+\cos x) (\cos^2 x + \sin^2 x) = (\sin x+\cos x)\cdot 1 = \sin x+\cos x[/tex]