`1/(cos(x) + 1) + 1/(cos(x) - 1) = -2csc(x)cot(x)` Verify the identity.

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Chapter 5, 5.2 - Problem 31 - Precalculus (3rd Edition, Ron Larson).
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mathace | (Level 3) Assistant Educator

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Verify the identity: `1/(cos(x)+1)+1/(cos(x)-1)=-2csc(x)cot(x)`

`[cos(x)-1+cos(x)+1]/(cos^2(x)-1)=-2csc(x)cot(x)`

`[2cos(x)]/(cos^2(x)-1)=-2csc(x)cot(x)`

Use the pythagorean identity `sin^2(x)+cos^2(x)=1` to simplify the denominator.

If `cos^2(x)-1`   is isolated the equation would be `cos^2(x)-1=-sin^2(x).`

`[2cos(x)]/[-sin^2(x)]=-2csc(x)cot(x)`

`[-2cos(x)]/[sin(x)sin(x)]=-2csc(x)cot(x)`

`-2*[1/sin(x)]*[cos(x)/sin(x)]=-2csc(x)cot(x)`

Use the reciprocal identity csc(x)=1/sin(x). Also use the quotient identity cot(x)=cos(x)/sin(x).

`-2csc(x)cot(x)=-2csc(x)cot(x)`

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balajia | College Teacher | (Level 1) eNoter

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By cross multiplication ,we get

`1/(cosx+1)+1/(cosx-1) = (cosx-1+cosx+1)/(cos^2x-1)`

`=(-2cosx)/(sin^2x)`

`=-2(cosx/sinx)(1/sinx)`

`=-2cotx.cscx`

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