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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 | 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:

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

` ` 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

`(2+2cosx)/(sinx(1+cosx))=(2(1+cosx))/(sinx(1+cosx))=2/sinx,`

which completes the proof.

` `

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