Math Help - Help Verifying an Identity

1. Help Verifying an Identity

I wasn't sure how to approach this problem, but I gave it a shot and now I'm stuck. Can someone please review what I have done.

Thanks

${csc{(2a)} - cot{(2a)} = tan{(a)}$
$={csc{(a+a)} - cot{(a+a)}$

$={csc{a} + csc{a} - cot{a} + cot{a}$

$={\frac{1}{\sin{a}}+ \frac{1}{sin{a}}} -{\frac{cos{a}}{\sin{a}}+ \frac{cos{a}}{sin{a}}}$

$={\frac{2}{\sin{a}} -{\frac{{2}cos{a}}{\sin{a}}$

2. Re: Help Verifying an Identity

Originally Posted by troe
I wasn't sure how to approach this problem, but I gave it a shot and now I'm stuck. Can someone please review what I have done.

Thanks
First of all, \displaystyle \begin{align*} \csc{2a} \neq \csc{a} + \csc{a} \end{align*}, and \displaystyle \begin{align*} \cot{2a} \neq \cot{a} + \cot{a} \end{align*}. Anyway

\displaystyle \begin{align*} \csc{2a} - \cot{2a} &= \frac{1}{\sin{2a}} - \frac{\cos{2a}}{\sin{2a}} \\ &= \frac{1 - \cos{2a}}{\sin{2a}} \\ &= \frac{1 - \left(1 - 2\sin^2{a}\right)}{2\sin{a}\cos{a}} \\ &= \frac{2\sin^2{a}}{2\sin{a}\cos{a}} \\ &= \frac{\sin{a}}{\cos{a}} \\ &= \tan{a} \end{align*}

3. Re: Help Verifying an Identity

Thank you very much for your help. I seem to have a problem with starting some identities and once I get started, I usually get stuck around the third step in the problem. Do you have any suggestions on the best approach in verifying identities? I know that I should begin with the most difficult side first, but I usually hit a road block once I begin the problem. Any tips, resources, or advice would be very helpful.

Thanks again.

4. Re: Help Verifying an Identity

Originally Posted by troe
Thank you very much for your help. I seem to have a problem with starting some identities and once I get started, I usually get stuck around the third step in the problem. Do you have any suggestions on the best approach in verifying identities? I know that I should begin with the most difficult side first, but I usually hit a road block once I begin the problem. Any tips, resources, or advice would be very helpful.

Thanks again.
Unless they're very simple, I usually convert everything to sines and cosines, because the sine and cosine identities are easy to remember and everything usually ends up easier to simplify.