Trigonometric Identity Problem

[Sabellic]

Homework Statement

http://img206.imageshack.us/img206/9099/titleol2.jpg
http://g.imageshack.us/g.php?h=206&i=titleol2.jpg

Show the above statement is equivalent to : sec (2x) + tan (2x)

Homework Equations

The Attempt at a Solution

First attempt in which I used the distributive Law early:
http://img237.imageshack.us/img237/876/attempt1pf3.jpg
http://g.imageshack.us/g.php?h=237&i=attempt1pf3.jpg

Second attempt in which I first solved within the brackets and then used the Distributive Law later:

http://img237.imageshack.us/img237/361/attempt2rz9.jpg
http://g.imageshack.us/g.php?h=237&i=attempt2rz9.jpg


Please help me.
Thanks.

 

[Geekchick]

Let see how do I put this...You have to add (multiply really) to your problem instead of dissecting it. After you do that you'll end up using the double angle formula on part of the top equation and the bottom. You will also have to use the pythagorean identity on the top. The main part however is the first step after you multiply it out the rest is rather straight forward. I don't know how good I am at giving directions so let me know if this makes any sense or not.

 

[Sabellic]

Thanks for the reply. Yes, it's very difficult to explain these online.

Where do you think I made my first error? which row of operations?

 

[Geekchick]

Well to start you keep trying to use the reciprocal forms and multiply that's not right. You have to leave the problem as is then multiply both the top and bottom by a pair of numbers that will end up, after using one of the identities mentions, a monomial denominator.
I don't think I am suppose to say exactly what to multiply it by since your not suppose to give straight answers on here. I can tell you all you use when solving it is sin and cos. How's that?

 

[Sabellic]

I tried with the conjugate. (cosx+sinx). But that didn't solve the problem.

Now i have:
1+sin2x / cos^2x - sin^2x

 

[Geekchick]

Thats nearly it all you have to do now is use the double angle formula on the bottom to get cos2x. Then just separate them so you have sin2x/cos2x + 1/cos2x then use the quotient identity and the reciprocal identity to get your answer. You should be able to see that.

 

[Defennder]

Convert them to the form of f(x) instead f(2x). Something will cancel out.

 

[Geekchick]

What do you mean?

 

[Sabellic]

Oh, I got it.!

Thank you very much.

 

[Geekchick]

oh you're welcome sabellic!

 

[Sabellic]

It's a matter of using the conjugate to get a denominator that is a monomial as Geekchick had said from the Double Angle Formulae. After simplifying, you need to use the reciprocal identities at the end.