Step by step
The question
Question: show tan x × cos x = sin 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 cos x on top has a partner at the bottom.
Each pair over itself is 1.
Cross them out.
What is left: sin x.
That is the right side, sin x.
So the identity is true!
3 Check
Let's check with a number: x = 30°.
On the calculator, tan 30° × cos 30° is about 0.5000.
And sin 30° is about 0.5000.
The same!
So it is right.
Let's look back.
First: write in sin and cos.
Then: cancel.
Last: check.