Sin^3x(1+Ctgx)+cos^3x(1+tgx)

Sin^3x(1+Ctgx)+cos^3x(1+tgx)

 

$$\\sin^3x(1+\frac{cosx}{sinx})+cos^3x(1+\frac{sinx}{cosx})\\\\
=sin^3x(\frac{sinx+cosx}{sinx})+cos^3x(\frac{cosx+sinx}{cosx})\\\\
=sin^2x(\frac{sinx+cosx}{1})+cos^2x(\frac{cosx+sinx}{1})\\\\
=(sinx+cosx)(sin^2x+cos^2x)\\\\
=(sinx+cosx)\times 1\\\\
=sinx+cosx\\\\$$