Prove that sin y*cos (x - y) + cos y*sin(x - y) = sin x

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justaguide | College Teacher | (Level 2) Distinguished Educator

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The identity `sin y*cos (x - y) + cos y*sin(x - y) = sin x` has to be proved.

Use the relation `sin a*cos b = (1/2)(sin (a - b) - sin(a + b))`

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

=> `(1/2)(sin (y - x + y) + sin(y + x - y)) + (1/2)*sin(x - y - y) + sin(x - y + y))`

=> `(1/2)(sin (2y - x) + sin x + sin(x - 2y) + sin x)`

=> `(1/2)(sin (2y - x) + sin x - sin(2y - x) + sin x)`

=> `(1/2)(sin x + sin x)`

=> `sin x`

This provesĀ  that ` sin y*cos (x - y) + cos y*sin(x - y) = sin x`

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