Question

Quem puder me ajudar, por favor, preciso de resolver esta questão: Sendo α + β = π/4 determine o valor de ( cos α + cos β )^2 + ( Sen α - Sen β )^2

Answer 1

Expandindo os quadrados:

$(\cos \alpha+\cos \beta)^2+(\mathrm{sen\,}\alpha-\mathrm{sen\,}\beta)^2\\\\ =\cos^2 \alpha+2\cos \alpha\cos \beta +\cos^2\beta+\mathrm{sen^2\,}\alpha-2\,\mathrm{sen\,}\alpha\,\mathrm{sen\,}\beta+\mathrm{sen^2\,}\beta\\\\ =(\cos^2\alpha+\mathrm{sen^2\,}\alpha)+(\cos^2\beta+\mathrm{sen^2\,}\beta)+2\cos \alpha\cos \beta-2\,\mathrm{sen\,}\alpha\,\mathrm{sen\,}\beta\\\\ =1+1+2\,(\cos \alpha\cos \beta-\mathrm{sen\,}\alpha\,\mathrm{sen\,}\beta)\\\\ =2+2\cos(\alpha+\beta)\\\\ =2+2\cos \dfrac{\pi}{4}\\\\\\ =2+2\cdot \dfrac{\sqrt{2}}{2}\\\\\\ =2+\sqrt{2}$