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Prove the following identity: `(1 + cos(x))/sin(x) + (sin(x))/(1+cos(x)) = (2)/(sinx)`

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foxwit | Student, Undergraduate | (Level 1) Salutatorian

Posted July 27, 2013 at 7:28 PM via web

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Prove the following identity:

`(1 + cos(x))/sin(x) + (sin(x))/(1+cos(x)) = (2)/(sinx)`

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degeneratecircle | High School Teacher | (Level 2) Associate Educator

Posted July 28, 2013 at 1:56 AM (Answer #1)

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We'll start from the left side and manipulate it until we get the right side. First, we get a common denominator as follows:


` ` Now `(1+cosx)^2=1+2cosx+cos^2x,` and if we add the fractions we get

`(1+cosx)^2/(sinx(1+cosx))+(sin^2x)/(sinx(1+cosx))=(1+2cosx+cos^2x+sin^2x)/(sinx(1+cosx))` .

But `cos^2x+sin^2x=1,` so the last fraction is equal to


which completes the proof.

` `


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