Answer:
Final answer is [tex]-\frac{\sqrt{2-\sqrt{3} }}{2}[/tex]
Step-by-step explanation:
Half angle formula is [tex]cos \frac{\alpha }{2} =[/tex] ±[tex]\sqrt{\frac{1+cos \alpha }{2} }[/tex]
First, since 105° is the 2nd quadrant, cosine is negative, so by the half angle formula above,
[tex]cos 150 =cos \frac{210}{2} = -\sqrt{\frac{1+cos(210)}{2} }[/tex]
[tex]cos 210 = cos(270-60)=-sin 60=-\frac{\sqrt{3} }{2}[/tex]
[tex]cos 105= -\sqrt{\frac{1-\frac{\sqrt{3} }{2} }{2} }=-\sqrt{\frac{2-\sqrt{3} }{4} } =-\frac{\sqrt{2-\sqrt{3} } }{2}[/tex]