How do I write $\csc$ and $\tan$ in terms of $\sin$ and $\cos$?

Question

I want to write $\csc$ and $\tan$ and terms of classical trigonometric functions like $\sin$ and $\cos$. I know about the identity $\sin(x)^2+\cos(x)^2=1$. But I am not sure where to go from here.

Answer 1

$$\tan x=\frac {\sin x}{\cos x}=\frac {\pm \sqrt{1-\cos^2x}}{\cos x}$$ for example. There a number of forms. Which one you want depends on what you want to do with it.

Answer 2

You have the following identity relating the cosecant to the sine: $$\csc(x)=\frac{1}{\sin(x)}$$

Similarly, you have

$$\tan(x)=\frac{\sin(x)}{\cos(x)}$$

relating the tangent to sine and cosine. These are the classics, but since the trigonometric functions all have interesting relationships among each other, you may transform these to various other identities.

Answer 3

$\csc x = \frac{1}{\sin x}$ and $\tan x = \frac{\sin x}{\cos x}$ by definition.

Answer 4

$$ \csc x= \frac{1}{\sin x}$$ or $$ \csc x= \frac{1}{\pm\sqrt{1-cos^2 x}}$$