Simplifying Trig Expressions Help?

Question

fanduekisses · Answer

$$\frac{ \cos ^{2}\theta }{\sin ^{2}\theta } + \csc \theta \sin \theta$$

fanduekisses · Answer

I'm so confused, I  got cot^2 but that is wrong

mathstudent55 · Answer

$ \dfrac{ \cos ^{2}\theta }{\sin ^{2}\theta } + \csc \theta \sin \theta$

$ = \cot ^{2}\theta  + \dfrac{1}{\sin \theta}\times  \sin \theta $

$ = \cot ^{2} \theta  + 1 $

fanduekisses · Answer

ooohh the pythagoren identity~ ! :D

fanduekisses · Answer

so it = csc^2 theta?

mathstudent55 · Answer

Now look at this:

$sin^2 \theta + \cos ^2 \theta = 1$

$ \dfrac{\sin^2 \theta}{sin^2 \theta} + \dfrac{\cos ^2 \theta}{\sin^2 \theta} = \dfrac{1}{\sin^2 \theta} $

$1 + \cot^2 \theta = \csc^2 \theta $

Therefore, you can replace $\cot^2 \theta\ + 1 $ with $\csc^2 \theta$

mathstudent55 · Answer

Yes, you got it!

fanduekisses · Answer

Thanks so much :)

mathstudent55 · Answer

You're welcome.