Prove the identity
sin (3 pi/2 + x) + sin (3 pi/2 -x) = -2 cos x
Use the identities
sin (a + b) = sin a cos b + sin b cos a
sin (a - b) = sin a cos b - sin b cos a
and the facts that sin 3 pi/2 = -1 and cos 3 pi/2 = 0
To prove the given identity, we need to simplify both sides of the equation and see if they are equal.
Let's start by simplifying the left-hand side (LHS) of the equation:
sin (3π/2 + x) + sin (3π/2 - x)
Using the sum formula for sine, we can rewrite sin (3π/2 + x) as:
sin (3π/2) cos (x) + cos (3π/2) sin (x)
The value of sin (3π/2) is -1, and cos (3π/2) is 0, so the expression becomes:
-1 * cos (x) + 0 * sin (x)
Which simplifies to just -cos (x).
Now let's simplify the right-hand side (RHS) of the equation:
-2 * cos(x)
Since the LHS simplifies to -cos(x) and the RHS is already -2cos(x), we can see that both sides are equal.
Thus, we have proved the identity sin (3π/2 + x) + sin (3π/2 - x) = -2cos(x).