Question

Sa se aduca la forma mai simpla expresia: (cos36°+sin²18°)÷cos²18°

Answer 1

$\displaystyle \cos 36^{\circ} = \cos (2 \cdot 18^{\circ})= 2 \cos^2 18^{\circ}-1 \\ \\ \frac{ \cos 36^{\circ}+ \sin^2 18^{\circ}}{ \cos^2 18^{\circ}}= \\ \\ =\frac{2 \cos ^2 18^{\circ}-1+ \sin ^2 18^{\circ}}{\cos^2 18^{\circ}}= \\ \\ =\frac{(\sin^2 18^{\circ}+\cos^2 18^{\circ})+\cos^2 18^{\circ}-1}{\cos^2 18^{\circ}}= \\ \\ =\frac{1+ \cos ^2 18^{\circ}-1}{ \cos^2 18^{\circ}}= \\ \\ =\frac{\cos^2 18^{\circ}}{\cos^2 18^{\circ}}= \\ \\ =1.$

$\displaystyle Identitati~folosite: \\ \\ \bullet \cos (2x)=2 \cos ^2 x-1 \\ \\ \bullet~\sin^2x+\cos^2x=1$

Answer 2

cos36°=cos (2*18)=cos²18-sin²18
(cos²18°-sin²18+sin²18)/cos²18°=cos²18/cos²18°=1