How do you verify the identity $sqrt((1-costheta)/(1+costheta))=(1-costheta)/abssintheta$ ?

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Answer 1 · 2017-01-19T07:39:03

All you really have to do is multiply by $sqrt(1 - costheta)/sqrt(1 - costheta)$ :

$=> sqrt((1 - costheta)/(1 + costheta))sqrt(1 - costheta)/sqrt(1 - costheta)$

$= (1 - costheta)/(sqrt((1 + costheta)(1 - costheta)))$

From the difference of two squares, we get:

$= (1 - costheta)/(sqrt(1 - cos^2theta))$

From the identity $sin^2theta + cos^2theta = 1$ , we get:

$= (1 - costheta)/(sqrt(sin^2theta))$

From the definition $sqrt(u^2) = |u|$ , we get:

$= color(blue)((1 - costheta)/|sintheta|)$ $color(blue)(sqrt"")$

since the square root of $u^2$ is necessarily positive.

How do you verify the identity sqrt((1-costheta)/(1+costheta))=(1-costheta)/abssinthetasqrt((1-costheta)/(1+costheta))=(1-costheta)/abssintheta?