Question

sin³x(1+tgx) +cos³x( 1+tgx)=sinx +cosx
Trebuie rezolvata/demonstrata egaliatea. Multumesc!!

Answer 1

   
[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]