simplify the expression sin(theta+pi/4) + sin(theta-pi/4)
To simplify the expression sin(theta + pi/4) + sin(theta - pi/4), we can use the trigonometric identity known as the sum-to-product identity:
sin(A) + sin(B) = 2 * sin((A + B)/2) * cos((A - B)/2)
So, let's apply this identity to the given expression:
sin(theta + pi/4) + sin(theta - pi/4)
= 2 * sin((theta + pi/4 + theta - pi/4)/2) * cos((theta + pi/4 - (theta - pi/4))/2)
= 2 * sin((2theta)/2) * cos(pi/2)
= 2 * sin(theta) * cos(pi/2)
Now, we know that cos(pi/2) equals 0, so we can simplify the expression further:
2 * sin(theta) * cos(pi/2) = 2 * sin(theta) * 0 = 0
Therefore, the simplified expression is 0.