Step by step
The question
Question: show cot x over cos x = csc x.
An identity is true for every angle x.
We work on the left side until it becomes the right side.
1 Write in sin and cos
Everything is easier in sin and cos.
cot is cos over sin.
We put them in.
Tops go on top, bottoms at the bottom.
2 Cancel
Now we look for the same thing on top and at the bottom.
Careful!
sin x means the sine of x.
It is one thing.
We cross out a whole sin x, never the x alone.
Every cos x on top has a partner at the bottom.
Each pair over itself is 1.
Cross them out.
What is left: 1 over sin x.
3 Name it
1 over sin x has its own name: csc x.
That is the right side, csc x.
So the identity is true!
4 Check
Let's check with a number: x = 30°.
On the calculator, cot 30° over cos 30° is about 2.0000.
And csc 30° is about 2.0000.
The same!
So it is right.
Let's look back.
First: write in sin and cos.
Then: cancel.
Then: name it.
Last: check.