Step by step
The question
Question: simplify (sec² x − 1) over tan x.
Simplify means: write it as simply as we can.
1 Use sin² + cos² = 1
The most important identity: sin² x + cos² x = 1.
It comes from the unit circle.
Divide every part by cos² x: tan² x + 1 = sec² x.
Moved around, it says sec² x − 1 = tan² x.
Careful!
The square stays: sec² x − 1 is tan² x, with the square.
So we put tan² x in place of (sec² x − 1).
2 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.
3 Cancel
A square means times itself: sin² x = sin x × sin x.
We write every one out.
Every sin x and cos x on top has a partner at the bottom.
Each pair over itself is 1.
Cross them out.
What is left: sin x over cos x.
4 Name it
sin x over cos x has its own name: tan x.
So it simplifies to tan x.
5 Check
Let's check with a number: x = 30°.
On the calculator, (sec² 30° − 1) over tan 30° is about 0.5774.
And tan 30° is about 0.5774.
The same!
So it is right.
Let's look back.
First: use sin² + cos² = 1.
Then: write in sin and cos.
Then: cancel.
Then: name it.
Last: check.