How do you prove #1/(1-cos)-1/(1+cos)= 2csc^2#?

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Answer 1 · 2015-05-17T19:04:27

The #-# sign in the question should be a #+#.

#1/(1-cos theta)+1/(1+cos theta)#

#= (1/(1-cos theta))((1+cos theta)/(1+cos theta)) + (1/(1+cos theta))((1-cos theta)/(1-cos theta))#

#= (1+cos theta)/(1-cos^2 theta) + (1-cos theta)/(1-cos^2 theta)#

#= ((1+cos theta) + (1-cos theta))/(1-cos^2 theta)#

#= 2/(sin^2 theta)#

#= 2(1/sin^2 theta)#

#= 2(1/sin theta)^2#

#= 2csc^2 theta#

If you try the same thing with #1/(1-cos theta)-1/(1+cos theta)# you will get the result #2 cos theta csc^2 theta#.