`(cos(x) - cos(y))/(sin(x) + sin(y)) + (sin(x) - sin(y))/(cos(x) + cos(y)) = 0` Verfiy the identity.

Expert Answers
Borys Shumyatskiy eNotes educator| Certified Educator

We'll use the formula `(a-b)*(a+b) = a^2-b^2.`

Multiply equation by `(sin(x)+sin(y))*(cos(x)+cos(y))`  (the product of the denominators):

`(cos(x)+cos(y))*(cos(x)-cos(y)) +(sin(x)+sin(y))*(sin(x)-sin(y)) = 0,`

`(cos^2(x)-cos^2(y)) + (sin^2(x)-sin^2(y)) = 0,`

`(cos^2(x)+sin^2(x)) - (cos^2(y)+sin^2(y)) = 0,`

1-1 = 0, QED.

embizze eNotes educator| Certified Educator

By multiplying both sides of the equation by the product of the denominators, you are assuming the equation is true. But that is what you are trying to prove.

Better is to rewrite the left side as a single fraction and note that the numerator is 0 (1-1) and the denominator is non-zero thus establishing the identity.

balajia | Student

By cross multiplication ,we get

`((cosx-coy)(cosx+cosy))+((sinx-siny)(sinx+siny))/((sinx+siny)(cosx+cosy))=`

`(cos^2x-cos^2y+sin^2x-sin^2y)/((sinx+siny)(cosx+cosy))=((cos^2x+sin^2x)-(cos^2y+sin^2y))/((sinx+siny)(cosx+cosy))`

`=(1-1)/((sinx+siny)(cosx+cosy))=0.`