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.