# Examine if equation sin(x-y)cosx-sinx cos(x-y) can be simplifyed?

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### 1 Answer

The given expression represents the expansion of the following trigonometric difference identity, such that:

`sin (theta - beta) = sin theta*cos beta - sin beta*cos theta`

Replacing `x - y` for `theta` and `x` for `beta` , yields:

`sin (x - y)*cos x - sin x*cos (x - y) = sin (x - y - x)`

Reducing duplicate angles, yields:

`sin (x - y)*cos x - sin x*cos (x - y) = sin(-y)`

By definition, `sin(-y) = -sin y` , hence, replacing `- sin y` for `sin(-y)` , yields:

`sin (x - y)*cos x - sin x*cos (x - y) = - sin y`

**Hence, reducing the given trigonometric expression, yields **`sin (x - y)*cos x - sin x*cos (x - y) = - sin y.`

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