Step by step
The question
Question: show (1 − cos² x) over sin x × cos x = tan x.
An identity is true for every angle x.
We work on the left side until it becomes the right side.
1 Use sin² + cos² = 1
The most important identity: sin² x + cos² x = 1.
It comes from the unit circle.
Moved around, it says 1 − cos² x = sin² x.
Careful!
The square stays: 1 − cos² x is sin² x, with the square.
So we put sin² x in place of (1 − cos² x).
2 Cancel
A square means times itself: sin² x = sin x × sin x.
We write every one out.
Every sin 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.
3 Name it
sin x over cos x has its own name: tan x.
That is the right side, tan x.
So the identity is true!
4 Check
Let's check with a number: x = 30°.
On the calculator, (1 − cos² 30°) over sin 30° × cos 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: cancel.
Then: name it.
Last: check.