`sqrt((1 + sin(theta))/(1 - sin(theta))) = (1 + sin(theta))/|cos(theta)|` Verfiy the identity.

Textbook Question

Chapter 5, 5.2 - Problem 41 - Precalculus (3rd Edition, Ron Larson).
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embizze's profile pic

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

Posted on

Verify `sqrt((1+sin(theta))/(1-sin(theta)))=(1+sin(theta))/|cos(theta)|`

Working from the left side, we show that it is equivalent to the right side:

`sqrt((1+sin(theta))/(1-sin(theta)))`

`=sqrt(1+sin(theta))/sqrt(1-sin(theta))`

`=sqrt(1+sin(theta))/sqrt(1-sin(theta))*sqrt(1+sin(theta))/sqrt(1+sin(theta))`

`=(1+sin(theta))/sqrt(1-sin^2(theta))`

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

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

balajia's profile pic

balajia | College Teacher | (Level 1) eNoter

Posted on

`L.H.S=sqrt((1+sin(theta))/(1-sin(theta)))`

Multiplying numerator and denominator with `1+sin(theta) ` ,we get

`sqrt(((1+sin(theta))((1+sin(theta)))/((1-sin(theta))(1+sin(theta)))`

`=sqrt((1+sin(theta))^2/cos^2(theta)) = (1+sin(theta))/|cos(theta)|.`

Because `1+sin(theta)` is always greater than or equal to zero.

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