Question

Can someone help me solve this problem involving trig identities? Problem in comments.

Answer 1

\[\frac{ \cos^2 \theta }{ \sin^2 \theta } + \csc \theta \sin \theta\]

Answer 2

Let's write what each of these means when finding sides of a right triangle.
\(\color{#000000}{ \displaystyle \sin \theta =({\rm opposite / hypotenuse}) }\)
\(\color{#000000}{ \displaystyle \cos \theta =({\rm adjacent / hypotenuse}) }\)

And you know that:
\(\color{#000000}{ \displaystyle \tan \theta = \frac{\rm opposite}{\rm adjacent} }\)

Therefore, when you divide top and bottom by adjacent you get:
\(\color{#000000}{ \displaystyle \tan \theta = \frac{({\rm opposite / hypotenuse})}{({\rm adjacent / hypotenuse})} }\)

And the substitution yields,
\(\color{#000000}{ \displaystyle \tan \theta = \frac{ \sin \theta}{\cos \theta} }\)

Answer 3

And the \(\color{#000000}{ \displaystyle \csc \theta = \frac{ 1}{\sin \theta} }\), because the \(\color{#000000}{ \displaystyle \csc\theta }\) is the reciprocal of the \(\color{#000000}{ \displaystyle \sin \theta }\).

Answer 4

You can further imply that:

\(\color{#000000}{ \displaystyle \tan \theta = \frac{ \sin \theta}{\cos \theta} }\)

\(\color{#000000}{ \displaystyle \left(\tan \theta \right)^2= \left(\frac{ \sin \theta}{\cos \theta} \right)^2 }\)

\(\color{#000000}{ \displaystyle \tan^2\theta= \frac{ \sin^2\theta}{\cos^2 \theta} }\)

Answer 5

And, \(\color{#000000}{ \displaystyle \sin\theta\times \csc\theta }\)
is just a product of a number and its reciprocal, which you should equal 1.

Answer 6

Then, you will need another rule:
\(\color{#000000}{ \displaystyle \tan^2\theta+1=\sec^2\theta }\)
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Answer 7

I was thinking that because it is cos/sin it would be equal to cot because cot = cos/sin
Am I wrong?

Answer 8

So then \[\cot^2 + 1 = \csc^2\] which is what wolfram alpha says it is.

Answer 9

Regardless if you or I had a faux pas I figured it out with your help. Thank you very much!

Answer 10

No, you are totally correct.

Answer 11

Nice site to refer to :)
In fact, I am using Mathematica program for some of tutorials/projects, and that is also really useful for everything in math like matrix reduction or almost anything yo can think of.

Answer 12

yw