[tex]\displaystyle\\
\text{Consider ca trebuia sa scrii:}\\
\sin^3x(1+\text{ctg }x) +\cos^3x( 1+\text{tg }x)=\sin x +\cos x\\\\
\text{\bf Rezolvare:}\\\\
\sin^3x(1+\text{ctg }x) +\cos^3x( 1+\text{tg }x)=\\\\
=\sin^3x \Big(1+ \frac{\cos x}{\sin x}\Big) +\cos^3x\Big(1+ \frac{\sin x}{\cos x}\Big)=\\\\
=\sin^3x \Big(\frac{\sin x+\cos x}{\sin x}\Big) +\cos^3x\Big(\frac{\cos x+\sin x}{\cos x}\Big)=\\\\
=\sin^3x \cdot \frac{\sin x+\cos x}{\sin x} +\cos^3x\cdot \frac{\cos x+\sin x}{\cos x}=
[/tex]
[tex]\displaystyle\\
=\frac{\sin^3x(\sin x+\cos x)}{\sin x} +\frac{\cos^3x (\cos x+\sin x)}{\cos x}=\\\\
=\sin^2x(\sin x+\cos x)+\cos^2x (\cos x+\sin x)=\\\\
=\sin^2x(\sin x+\cos x)+\cos^2x (\sin x+\cos x)=\\\\
(\sin x+\cos x)(\sin^2x+\cos^2x)=\boxed{\sin x+\cos x}\\\\
\text{cctd}[/tex]
