Find the exact value of sin157.5 in half angle identities

We can find the exact value of sin 157.5 like this:

sin²(157.5) = 1/2 (1 − cos(2*157.5)) 
                = 1/2 (1 − cos(315)) 
                = 1/2 (1 − √2/2) 
                = (2−√2)/4 

Since 157.5 is in quadrant II, then sin(157.5) > 0, so we take positive square root. 
sin(157.5) = √(2−√2)/2 
I really hope this can help you