Question

using angle sum identities, how does cos 165 = sq root (2 + sq root 3)/2?

Answer 1

cos(165)=cos(120+45)
=cos120cos45-sin120sin45
=cos(90+30)cos45-sin(90+30)sin45
Now,
cos(90+30)=-sin30
sin(90+30)=cos30
And values of cos45 and sin45 are equal so we can write cos45=sin45 and take a common in the equation say..cos45
=cos45(-sin30-cos30)
=-cos45(sin30+cos30)
Putting in values we get the value of the exp to be=\[-\frac{ 1 }{ \sqrt{2}}(\frac{ 1 }{ 2}+\frac{ \sqrt{3} }{ 2})\]

Nowhere near the proof..have I made any mistake?