Prove Trig Identities

[speeddown]

I've tried searching around and haven't been able to find anything on this one in particular. I've been working on it for about an hour now:



___2___ = sin2x
cotx + tanx


I haven't had much trouble with proving identities till now.

 

[speeddown]

Here's where I get stuck

\(\displaystyle =\dfrac{2}{\dfrac{\cos \left(x\right)}{\sin \left(x\right)}\,+\,\dfrac{\sin \left(x\right)}{\cos \left(x\right)}}\)

I can get there, easy

But I'm not sure how they get the following:

\(\displaystyle =\dfrac{2\cos \left(x\right)\sin \left(x\right)}{\cos ^2\left(x\right)\,+\,\sin ^2\left(x\right)}\)

 

[stapel]

\(\displaystyle =\dfrac{2}{\dfrac{\cos \left(x\right)}{\sin \left(x\right)}\,+\,\dfrac{\sin \left(x\right)}{\cos \left(x\right)}}\)

I can get there, easy

But I'm not sure how they get the following:

\(\displaystyle =\dfrac{2\cos \left(x\right)\sin \left(x\right)}{\cos ^2\left(x\right)\,+\,\sin ^2\left(x\right)}\)

Don't forget what you learned about dividing by a fraction!

What did you get when you converted the terms in the denominator of the first line above to a common denominator, combined into one fraction, and then flipped and multiplied?

 

[Deleted member 4993]

\(\displaystyle =\dfrac{2}{\dfrac{\cos \left(x\right)}{\sin \left(x\right)}\,+\,\dfrac{\sin \left(x\right)}{\cos \left(x\right)}}\)

I can get there, easy

But I'm not sure how they get the following:

\(\displaystyle =\dfrac{2\cos \left(x\right)\sin \left(x\right)}{\cos ^2\left(x\right)\,+\,\sin ^2\left(x\right)}\)

Without using calculator - can you calculate (and express the sum as a fraction):

\(\displaystyle \dfrac{2}{\dfrac{3}{4}\,+\,\dfrac{5}{7}} \ = \ ??\)