Como a dica diz, 105º = 45º+60º, logo [tex]\cos(105^\circ)=\cos(45^\circ+60^\circ)[/tex]. Sabe-se que [tex]\cos(a+b)=\cos a\cos b-\sin a\sin b[/tex], logo:
[tex]\cos(105^\circ)=\cos45^\circ\cos60^\circ-\sin45^\circ\sin60^\circ[/tex]
[tex]\cos(105^\circ)=\frac{\sqrt{2}}{2}\cdot\frac{1}{2}-\frac{\sqrt{2}}{2}\cdot\frac{\sqrt{3}}{2}[/tex]
[tex]\cos(105^\circ)=\frac{\sqrt{2}}{4}-\frac{\sqrt{2\cdot3}}{4}[/tex]
[tex]\cos(105^\circ)=\frac{\sqrt{2}}{4}-\frac{\sqrt{6}}{4}[/tex]
[tex]\cos(105^\circ)=\frac{\sqrt{2}-\sqrt{6}}{4}[/tex]
