# Thread: Showing A Identity Holds

1. ## Showing A Identity Holds

I have to show that the following identity holds when 1 + cos(2Ө) does not equal 0.

tan(Ө) = sin(2Ө)/1 + cos(2Ө)

I think to do this, you'd have to use cos2Ө and sin2Ө's formulas to derive a formula for tan(Ө) or tan(2Ө). It's the (2Ө) thing that throws me off.

2. Originally Posted by SportfreundeKeaneKent
I have to show that the following identity holds when 1 + cos(2Ө) does not equal 0.

tan(Ө) = sin(2Ө)/1 + cos(2Ө)

I think to do this, you'd have to use cos2Ө and sin2Ө's formulas to derive a formula for tan(Ө) or tan(2Ө). It's the (2Ө) thing that throws me off.
$tan(\theta) = \frac{sin(2\theta)}{1+cos(2\theta)}$

$= \frac{2~sin(\theta)~cos(\theta)}{1+cos(2\theta)}$

$= \frac{2~sin(\theta)~cos(\theta)}{1+cos^2(\theta)-sin^2(\theta)}$

$= \frac{2~sin(\theta)~cos(\theta)}{2cos^2(\theta)}$

$= \frac{sin(\theta)}{cos(\theta)}$

$= tan(\theta)$

3. Thanks. The third line is tricky.

I wanna get a forumula for tan(2Ө) in terms of tan(Ө) by using the cos(2Ө) = cos^2(Ө) - sin^2(Ө) and sin(2Ө) = 2sinӨcosӨ formulas.

Can I use tan(2Ө) = sin2Ө/cos2Ө and go on from there? And how would I get it back to tanӨ?

4. Hello,

Originally Posted by SportfreundeKeaneKent
Thanks. The third line is tricky.

I wanna get a forumula for tan(2Ө) in terms of tan(Ө) by using the cos(2Ө) = cos^2(Ө) - sin^2(Ө) and sin(2Ө) = 2sinӨcosӨ formulas.

Can I use tan(2Ө) = sin2Ө/cos2Ө and go on from there? And how would I get it back to tanӨ?
By dividing if necessary

$\tan 2 \theta=\frac{2 \sin \theta \cos \theta}{\cos^2 \theta-\sin^2 \theta}$

Divide by $\cos^2 \theta$ above and below

5. I can get

(2sinӨ/cosӨ)/1-tan^2Ө

2tanӨ/1-tan^2Ө = tan(2Ө)

Would that be the formula or could you simplify it more?

6. Originally Posted by SportfreundeKeaneKent
I can get

(2sinӨ/cosӨ)/1-tan^2Ө

2tanӨ/1-tan^2Ө = tan(2Ө)

Would that be the formula or could you simplify it more?
It's the formula