# Thread: Proving an identity that's proving to be complex

1. ## Proving an identity that's proving to be complex

Prove the identity cot A/2 - tan A/2 = 2cot A

I have written it as:

$\displaystyle \frac{cos \frac{A}{2}}{sin \frac{A}{2}} - \frac { sin \frac {A}{2}}{cos \frac{A}{2}}$

Taking LCM:

$\displaystyle \frac {cos^2 \frac{A}{2} - sin^2 \frac{A}{2}}{sin \frac{A}{2} cos \frac{A}{2}}$

Now I don't know how to solve it any further?

Is there any easier method of proving this identity ?

2. Originally Posted by struck
Prove the identity cot A/2 - tan A/2 = 2cot A

I have written it as:

$\displaystyle \frac{cos \frac{A}{2}}{sin \frac{A}{2}} - \frac { sin \frac {A}{2}}{cos \frac{A}{2}}$

Taking LCM:

$\displaystyle \frac {cos^2 \frac{A}{2} - sin^2 \frac{A}{2}}{sin \frac{A}{2} cos \frac{A}{2}}$

Now I don't know how to solve it any further?

Is there any easier method of proving this identity ?
I'd do it your way too and you were only two steps away from the end. Carrying on:

For the numerator recall that $\displaystyle cos(2u) = cos^2(u) - sin^2(u)$

For the denominator recall that $\displaystyle sin(2u) = 2sin(u)cos(u) \rightarrow cos(u)sin(u) = \frac{1}{2}sin(2u)$

This will reduce your equation to $\displaystyle \frac{cos(2u)}{\frac{1}{2}sin(2u)}$ which I'm sure you can simplify to get the answer