# Trig Identity proof

1. Apr 6, 2013

### trollcast

1. The problem statement, all variables and given/known data

Prove the identity:

$$\csc(2\theta)-\cot(2\theta)\equiv\tan(\theta)$$

2. Relevant equations

3. The attempt at a solution

Starting with the LHS:

$$\csc(2\theta)-\cot(2\theta)$$
$$\frac{1}{\sin(2\theta)}-\frac{\cos(2\theta)}{\sin(2\theta)}$$
$$\frac{1-\cos(2\theta)}{sin(2\theta)}$$

And thats as far as I can see to rearrange it.

2. Apr 6, 2013

### rock.freak667

Now just substitute your identities for cos2θ and sin2θ and it should work out easily.

3. Apr 6, 2013

### trollcast

Woops I forgot about the double angle formulae

so the rest of it is:

$$\frac{1-(1-2\sin^2(\theta))}{2sin(\theta)\cos(\theta)}$$
$$\frac{2\sin^2(\theta)}{2sin(\theta)\cos(\theta)}$$
$$\frac{sin(\theta)}{\cos(\theta)}$$
$$=\tan(\theta)$$
∴ LHS = RHS