`(1 + sin(theta))/cos(theta) + cos(theta)/(1 + sin(theta)) = 2sec(theta)` Verfiy the identity.

Textbook Question

Chapter 5, 5.2 - Problem 29 - Precalculus (3rd Edition, Ron Larson).
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Borys Shumyatskiy | College Teacher | (Level 3) Associate Educator

Posted on

Transform the left part:

`(1 + sin(theta))/cos(theta) + cos(theta)/(1 + sin(theta)) =`

`((1 + sin(theta))^2 + cos^2(theta))/(cos(theta)*(1 + sin(theta))) =`

`(1 + 2sin(theta) + sin^2(theta) + cos^2(theta))/(cos(theta)*(1 + sin(theta))) =`

`(2 + 2sin(theta))/(cos(theta)*(1 + sin(theta))) = 2/cos(theta) = 2sec(theta), ` QED.

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balajia | College Teacher | (Level 1) eNoter

Posted on

`(1+sin(theta))/cos(theta)+cos(theta)/(1+sin(theta)) = sec(theta)+tan(theta)+((cos(theta)(1-sin(theta)))/(1-sin^2(theta)))`

`=sec(theta)+tan(theta)+((cos(theta)(1-sin(theta)))/(cos^2(theta)))`

`=sec(theta)+tan(theta)+(1-sin(theta))/(cos(theta))`

`=sec(theta)+tan(theta)+sec(theta)-tan(theta)`

`=2sec(theta).`

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