Trig Identify Verification

[Peterson25]

Hello, Hopefully I can get some help with two questions before my test today!

I'm currently having issues with verifying an identify.

sqrt((1-cos(Theta))/(1+cos(Theta))) = (1-cos(Theta))/abs(sin(Theta))

Getting rid of the square root is easy enough, but I'm having issues going from 1+cos(Theta) to abs(sin(Theta)). I've considered a Pythagorean identity, but I'm not too positive that's the correct way.

Also, I have one other question. I need help with finding the exact value of cos(55deg)^2+cos(35deg)^2. The review says "Do not use a calculator. State which identities you have used." I've considered a reciprocal identify and the Pythagorean identity, but I'm not sure how to work the problem.

If you could please help me it would be greatly appreciated, and thanks in advance.

 

[galactus]

For your second problem, take note that cos(35)=sin(55). The answer should pop out at you upon knowing that. See?.

 

[pka]

\(\displaystyle \L\begin{array}{rcl}
\sqrt {\frac{{1 - \cos (\phi )}}{{1 + \cos (\phi )}}} & = & \sqrt {\frac{{1 - \cos (\phi )}}{{1 + \cos (\phi )}}} \sqrt {\frac{{1 - \cos (\phi )}}{{1 - \cos (\phi )}}} \\
& = & \frac{{1 - \cos (\phi )}}{{\sqrt {\sin ^2 (\phi )} }} \\
& = & \frac{{1 - \cos (\phi )}}{{\left| {\sin (\phi )} \right|}}\quad \mbox{recall that}\quad \sqrt {x^2 } = \left| x \right| \\
\end{array}\).

 

[Peterson25]

pka, that's fantastic! Thank you for helping me. May I ask what type of software you used to create the formula?

galactus, could you please elaborate? The answer isn't popping out at the moment. Perhaps it's because I've been up a good portion of the night trying to complete this work! I know the answer is 1, or atleast I think that's the correct solution, but how did you arrive to that conclusion, and which identities should I pay attention to?

Thanks guys, what a quick reply too!

 

[galactus]

pka didn't use software. He used Latex. Type in the code. Click on quote to see the code.

As for the identity, \(\displaystyle cos^{2}(55)+sin^{2}(55)=1\)

You don't see that?. It just comes from the ever popular \(\displaystyle cos^{2}(x)+sin^{2}(x)=1\)

 

[Peterson25]

Sorry, I didn't know about the the "Latex" code functions.

Yeah, I see it now. I had to take a breather for a few minutes. Anyway, Thanks for your help!

I'll probably be coming around here more often. It seems like a great community.

 

[galactus]

No need to be sorry about not knowing LaTex. I didn't either when I started here. But I learned.