# Math Help - Proving an Identity

1. ## Proving an Identity

Prove the identity cosecA-cotA = tan(0.5A)

2. Originally Posted by Punch
Prove the identity cosecA-cotA = tan(0.5A)

Try using half angle identities.

$\sin{\frac{\theta}{2}} = \sqrt{\frac{1-\cos{\theta}}{2}}$

and

$\cos{\frac{\theta}{2}} = \sqrt{\frac{1+\cos{\theta}}{2}}$

It should into an easy problem using these identities.

3. Originally Posted by 11rdc11
Try using half angle identities.

$\sin{\frac{\theta}{2}} = \sqrt{\frac{1-\cos{\theta}}{2}}$

and

$\cos{\frac{\theta}{2}} = \sqrt{\frac{1+\cos{\theta}}{2}}$

It should into an easy problem using these identities.
I haven't learnt any half angles identities or formula yet

4. Have you learnt double angle identities yet?

5. yes

6. Originally Posted by Punch
yes
Ok let me show you how to derive them from the double angle

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

$\cos{2\theta} = \cos^2{\theta} -(1-\cos^2{\theta})$

$\cos{2\theta} = \cos^2{\theta}-1 + \cos^2{\theta}$

$\cos{2\theta} = 2\cos^2{\theta} -1$

$\cos{2\theta} +1= 2\cos^2{\theta}$

$\frac{\cos{2\theta}+1}{2}= \cos^2{\theta}$

$\cos{\theta} = ^+_- \sqrt{\frac{\cos{2\theta}+1}{2}}$

now replace $\theta$ with $\frac{\theta}{2}$

$\cos{\frac{\theta}{2}} = ^+_- \sqrt{\frac{\cos{2(\frac{\theta}{2})}+1}{2}}$

which equals

$\cos{\frac{\theta}{2}} = ^+_- \sqrt{\frac{\cos{\theta}+1}{2}}$

Same idea to derive for sin

7. $\tan{\frac{\theta}{2}} = \frac{\sin{\frac{\theta}{2}}}{\cos{\frac{\theta}{2 }}} = \frac{\sqrt{\frac{1-\cos{\theta}}{2}}}{\sqrt{\frac{1+\cos{\theta}}{2}} } = \sqrt{\frac{1-\cos{\theta}}{1+\cos{\theta}}} = \sqrt{\frac{(1-\cos{\theta})(1-\cos{\theta})}{(1+\cos{\theta})(1-\cos{\theta)}}} = \sqrt{\frac{(1-\cos{\theta})^2}{(1-\cos^2{\theta})}}$

= $\frac{1-\cos{\theta}}{\sin{\theta}} = \frac{1}{\sin{\theta}} - \frac{\cos{\theta}}{\sin{\theta}} = \csc{\theta} - \cot{\theta}$

8. I don't really understand it...

Can you show me how to start and solve the question from

cosecA-cotA = tan(0.5A) without using half angle? sorry

9. Do as you have been told.

Use the double angle identities for sine and cosine to derive the half angle identities.

Then do $\frac{\sin{\frac{\theta}{2}}}{\cos{\frac{\theta}{2 }}}$.

10. Thanks guys, finally understood half angle, thankd rdc