Verify this identity in Trig?

Question

sin((3pi/2)+x)+sin((3pi/2)-x)=-2cosx

Hopefully that isn't confusing. Step by step, please!

Answer 1

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each bit on the LHS can be simplified first using co-function identities then put together to get the RHS

3π/2 + x is in quadrant 4 ... so sin [(3π/2) + x] = -cos x

and 3π/2 - x is in quadrant 3 ... so sin [(3π/2) - x] = -cos x

so sin [(3π/2) + x] + sin [(3π/2) - x] = -cos x - cos x

= -2cos x

edit:

btw ... just in case you're wondering why sin [(3π/2) + x] = -cos x and NOT cos x ... it's b/c sine is negative in quadrant 4 and that determines the sign of the co-function ... gotta be careful about that