These expressions look complicated, but they are not if you know your trigonometric identities. Review the addition/subtraction formulas, double-angle formulas, and half-angle formulas. Then simplify each of the expressions below.  

1. $\sin(50^\circ)\cos(20^\circ)+\cos(50^\circ)\sin(20^\circ)$ 

   Hint (a):

   Use the angle sum formula for sine.

2. $2\sin\left(\frac{2\pi}{5}\right)\cos\left(\frac{2\pi}{5}\right)$ 

   Hint (b):

   Use the Double-Angle formula for the sine.

   Step (b):

   $\text{sin}\left( 2\left( \frac{2\pi}{5} \right) \right)$

3. $\cos^2\left(\frac{\pi}{7}\right)-\sin^2\left(\frac{\pi}{7}\right)$ 

   Hint (c):

   Use the Double-Angle formula for the cosine.

4. $\cos(110^\circ)\cos(50^\circ)-\sin(110^\circ)\sin(50^\circ)$ 

   Hint (d):

   Use the angle sum formula for the cosine.

5. $\sqrt { \frac { 1 + \operatorname { cos } ( 20 ^ { \circ } ) } { 2 } }$ 

   Hint (e):

   Use the Half-Angle formula for cosine.

6. $\sqrt { \frac { 1 - \operatorname { cos } ( 20 ^ { \circ } ) } { 2 } }$ 

   Hint (f):

   Use the Half-Angle formula for sine.