Step by step
The question
Question: simplify sin x over tan x.
Simplify means: write it as simply as we can.
sin xtan x
1 Write in sin and cos
Everything is easier in sin and cos.
tan is sin over cos.
tan x = sin xcos x
We put them in.
Tops go on top, bottoms at the bottom.
= sin x × cos xsin x
2 Cancel
Now we look for the same thing on top and at the bottom.
= sin x × cos xsin x
Careful!
sin x means the sine of x.
It is one thing.
We cross out a whole sin x, never the x alone.
Careful: sin x is one thing
Every sin x on top has a partner at the bottom.
Each pair over itself is 1.
Cross them out.
What is left: cos x.
= cos x
So it simplifies to cos x.
3 Check
Let's check with a number: x = 30°.
On the calculator, sin 30° over tan 30° is about 0.8660.
x = 30°: sin 30°tan 30° ≈ 0.8660
And cos 30° is about 0.8660.
The same!
So it is right.
cos 30° ≈ 0.8660
Let's look back.
First: write in sin and cos.
Then: cancel.
Last: check.
Answer
= cos x