How do you prove that $tan(x/2)-cot(x/2) = -2cotx$ ?

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Answer 1 · 2016-02-23T09:59:58

$tan(x/2)−cot(x/2)=-2cotx$ For proof see below.

$tan(x/2)−cot(x/2)$

= $sin(x/2)/cos(x/2)-cos(x/2)/sin(x/2)$

= $(sin^2(x/2)-cos^2(x/2))/(sin(x/2)cos(x/2))$

As $cosx=cos^2(x/2)-sin^2(x/2)$ and $sinx=2sin(x/2)cos(x/2)$

Above can be written as $-cosx/(sinx/2)$ or

$-2cotx$

How do you prove that tan(x/2)-cot(x/2) = -2cotx tan(x/2)-cot(x/2) = -2cotx ?