Simplify
cos{A + (n-3/2)B} - cos{A+ (n+3/2)B}
To simplify the expression cos{A + (n-3/2)B} - cos{A+ (n+3/2)B}, we can use the formula for the difference of cosines:
cos(x) - cos(y) = -2 * sin((x+y)/2) * sin((x-y)/2)
Let's substitute x = A + (n-3/2)B and y = A + (n+3/2)B into the formula:
cos{A + (n-3/2)B} - cos{A+ (n+3/2)B} = -2 * sin((A + (n-3/2)B + A + (n+3/2)B)/2) * sin((A + (n-3/2)B - (A + (n+3/2)B))/2)
Simplifying this further:
= -2 * sin((2A + 2nB)/2) * sin((-2B)/2)
= -2 * sin(A + nB) * sin(-B)
Now, we can simplify sin(-B) using the property of sine function: sin(-x) = -sin(x):
= -2 * sin(A + nB) * (-sin(B))
= 2 * sin(A + nB) * sin(B)
Therefore, the simplified expression is 2 * sin(A + nB) * sin(B).