How do you determine the restrictions of an identity?

[eleventhxhour]

So, this question says "prove each identity. State any restrictions on the variables".

5a) \(\displaystyle \frac{sinx}{tanx} = cosx\)

I did the first part of the question correctly (proving it), but I don't understand how you determine the restrictions on the variables. In the textbook, it says that it cannot be 0°, 90°, 180°, 270°, and 360°. Could someone explain how they got that?

Thanks!

 

[MarkFL]

We cannot have division by zero, so we know:

\(\displaystyle \tan(x)\ne0\)

And because \(\displaystyle \tan(x)\equiv\frac{\sin(x)}{\cos(x)}\implies\sin(x)\ne0\)

We also have:

\(\displaystyle \cos(x)\ne0\)

 

[eleventhxhour]

We cannot have division by zero, so we know:

\(\displaystyle \tan(x)\ne0\)

And because \(\displaystyle \tan(x)\equiv\frac{\sin(x)}{\cos(x)}\implies\sin(x)\ne0\)

We also have:

\(\displaystyle \cos(x)\ne0\)

Okay, so that gets you that it cannot = 0° and 90°. But how did they get that it cannot be 180°, 270°, and 360°?

 

[MarkFL]

What are:

\(\displaystyle \sin\left(180^{\circ}\right)\)

\(\displaystyle \cos\left(270^{\circ}\right)\)

\(\displaystyle \sin\left(360^{\circ}\right)\)

 

[eleventhxhour]

What are:

\(\displaystyle \sin\left(180^{\circ}\right)\)

\(\displaystyle \cos\left(270^{\circ}\right)\)

\(\displaystyle \sin\left(360^{\circ}\right)\)

They all equal 0