trigonometry
How do you simplify trigonometric functions like $\tan(x)\sec(x)$ or $\csc(x)\cot(x)$ as well as other equations like $\frac{\tan(x)}{\sec(x)}$ and so on? And could you explain why you are doing the steps so I can understand it a little better and be able to do these on my own. What about $\sec(x)\cot(x)$?
Thank you.
Here is the Wikipedia page on the subject, and here is an image of a unit circle from that page that answers your question quite well. Credit for this image goes to Wikipedia User:Tttrung.
You made a mistake in one of your steps. You should have
$$\sin^4x + \cos^4x = (\sin^2x+\cos^2x)^2-2\sin^2x\cos^2x$$
This then helps you achieve
$$\sqrt{\sec^2x\csc^2x-2}$$
Other than that, none of the choices are correct (as far as I can tell, the above expression can't be simplified any further).
Best Answer
It pretty much boils down to using the definitions of $\tan x, \sec x, \csc x, \cot x$.
For example
$$\tan x \sec x = \dfrac{\sin x}{\cos x}\cdot \dfrac 1{\cos x} = \dfrac {\sin x}{\cos^2 x}$$
$$\csc x \cot x = \dfrac{1}{\sin x} \cdot \dfrac{\cos x}{\sin x} = \dfrac{\cos x}{\sin^2 x}$$
$$\sec x \cot x = \require{cancel} \dfrac 1{\cancel{\cos x}}\cdot \dfrac{\cancel{\cos x}}{\sin x} = \dfrac 1{\sin x} = \csc x$$