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

Textbook Question

Chapter 5, 5.2 - Problem 40 - Precalculus (3rd Edition, Ron Larson).
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shumbm's profile pic

Borys Shumyatskiy | College Teacher | (Level 3) Associate Educator

Posted on

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.

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embizze | High School Teacher | (Level 1) Educator Emeritus

Posted on

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.

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balajia | College Teacher | (Level 1) eNoter

Posted on

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.`

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