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

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

 

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