Proving Trig Identity: $\csc(2\theta)-\cot(2\theta)\equiv\tan(\theta)$

[trollcast]

Homework Statement

Prove the identity:

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

Homework Equations

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 that's as far as I can see to rearrange it.

 

[rock.freak667]

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

 

[trollcast]

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

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