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 $2 cos theta csc^2 theta$ .

How do you prove 1/(1-cos)-1/(1+cos)= 2csc^21/(1-cos)-1/(1+cos)= 2csc^2?