Step by step
The question
Question: simplify tan x over sin x.
Simplify means: write it as simply as we can.
tan xsin x
1 Write in sin and cos
Everything is easier in sin and cos.
tan is sin over cos.
tan x = sin xcos x
We put them in.
Tops go on top, bottoms at the bottom.
= sin xcos x × sin x
2 Cancel
Now we look for the same thing on top and at the bottom.
= sin xcos x × sin x
Careful!
sin x means the sine of x.
It is one thing.
We cross out a whole sin x, never the x alone.
Careful: sin x is one thing
Every sin x on top has a partner at the bottom.
Each pair over itself is 1.
Cross them out.
What is left: 1 over cos x.
= 1cos x
3 Name it
1 over cos x has its own name: sec x.
= sec x
So it simplifies to sec x.
4 Check
Let's check with a number: x = 30°.
On the calculator, tan 30° over sin 30° is about 1.1547.
x = 30°: tan 30°sin 30° ≈ 1.1547
And sec 30° is about 1.1547.
The same!
So it is right.
sec 30° ≈ 1.1547
Let's look back.
First: write in sin and cos.
Then: cancel.
Then: name it.
Last: check.
Answer
= sec x