MHB How do you determine the restrictions of an identity?

[eleventhxhour]

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

5a) $$\frac{sinx}{tanx} = cosx$$

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:

$$\tan(x)\ne0$$

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

We also have:

$$\cos(x)\ne0$$

 

[eleventhxhour]

We cannot have division by zero, so we know:

$$\tan(x)\ne0$$

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

We also have:

$$\cos(x)\ne0$$

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:

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

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

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

 

[eleventhxhour]

What are:

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

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

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

They all equal 0