Answer
$sin~75°=\frac{\sqrt 6+\sqrt 2}{4}$
$cos~75°=\frac{\sqrt 6-\sqrt 2}{4}$
$tan~75°=\frac{\sqrt 3+1}{\sqrt 3-1}$
Work Step by Step
$sin~75°=sin~(120°-45°)=sin~120°~cos~45°-cos~120°~sin~45°=\frac{\sqrt 3}{2}\frac{\sqrt 2}{2}-(-\frac{1}{2})\frac{\sqrt 2}{2}=\frac{\sqrt 6}{4}+\frac{\sqrt 2}{4}=\frac{\sqrt 6+\sqrt 2}{4}$
$cos~75°=cos~(120°-45°)=cos~120°~cos~45°+sin~120°~sin~45°=-\frac{1}{2}\frac{\sqrt 2}{2}+\frac{\sqrt 3}{2}\frac{\sqrt 2}{2}=-\frac{\sqrt 2}{4}+\frac{\sqrt 6}{4}=\frac{\sqrt 6-\sqrt 2}{4}$
$tan~75°=tan(120°-45°)=\frac{tan~120°-tan~45°}{1+tan~120°~tan~45°}=\frac{-\sqrt 3-1}{1+(-\sqrt 3)1}=\frac{-\sqrt 3-1}{1-\sqrt 3}=\frac{\sqrt 3+1}{\sqrt 3-1}$
