Answer:
[tex]\frac{1}{4}[/tex]( [tex]\sqrt{6}[/tex] - [tex]\sqrt{2}[/tex])
Step-by-step explanation:
Using the addition formula for cosine
cos(x - y) = cosxcosy + sinxsiny
and the exact values
sin45° = cos45° = [tex]\frac{1}{\sqrt{2} }[/tex]
cos60° = [tex]\frac{1}{2}[/tex], sin60° = [tex]\frac{\sqrt{3} }{2}[/tex]
Note that
cos(- 75)° = cos(45 - 120)°, thus
cos(45 - 120)°
= cos45° cos120° + sin45° sin120°
= cos45° ( - cos60°) + sin45° sin60°
= [tex]\frac{1}{\sqrt{2} }[/tex] × - [tex]\frac{1}{2}[/tex] + [tex]\frac{1}{\sqrt{2} }[/tex] × [tex]\frac{\sqrt{3} }{2}[/tex]
= - [tex]\frac{1}{2\sqrt{2} }[/tex] + [tex]\frac{\sqrt{3} }{2\sqrt{2} }[/tex]
= [tex]\frac{\sqrt{3}-1 }{2\sqrt{2} }[/tex] × [tex]\frac{\sqrt{2} }{\sqrt{2} }[/tex]
= [tex]\frac{\sqrt{2}(\sqrt{3}-1) }{4}[/tex]
= [tex]\frac{1}{4}[/tex]( [tex]\sqrt{6}-\sqrt{2}[/tex])
