Step by step
The question
Question: show tan x over sin x = sec 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.
tan is sin over cos.
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 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.
3 Name it
1 over cos x has its own name: sec x.
That is the right side, sec x.
So the identity is true!
4 Check
Let's check with a number: x = 30°.
On the calculator, tan 30° over sin 30° is about 1.1547.
And sec 30° is about 1.1547.
The same!
So it is right.
Let's look back.
First: write in sin and cos.
Then: cancel.
Then: name it.
Last: check.