Step by step
The question
Question: simplify (1 − cos² x) over sin x.
Simplify means: write it as simply as we can.
(1 − cos² x)sin x
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.
sin² x + cos² x = 1
so 1 − cos² x = sin² x
Careful!
The square stays: 1 − cos² x is sin² x, with the square.
Careful: not 1 − cos² x = sin x ✗
So we put sin² x in place of (1 − cos² x).
= sin2 xsin x
2 Cancel
A square means times itself: sin² x = sin x × sin x.
We write every one out.
= sin x × sin xsin x
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.
= sin x
So it simplifies to sin x.
3 Check
Let's check with a number: x = 30°.
On the calculator, (1 − cos² 30°) over sin 30° is about 0.5000.
x = 30°: (1 − cos² 30°)sin 30° ≈ 0.5000
And sin 30° is about 0.5000.
The same!
So it is right.
sin 30° ≈ 0.5000
Let's look back.
First: use sin² + cos² = 1.
Then: cancel.
Last: check.
Answer
= sin x