Step by step
The question
Question: simplify (1 + tan² x) × cos² 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 1 + tan² x = sec² x.
Careful!
The square stays: 1 + tan² x is sec² x, with the square.
So we put sec² x in place of (1 + tan² x).
2 Write in sin and cos
Everything is easier in sin and cos.
sec is 1 over cos.
We put them in.
Tops go on top, bottoms at the bottom.
3 Cancel
Now we look for the same thing on top and at the bottom.
cos x, squared, is on top and at the bottom.
Over itself it is 1.
Cross them out.
What is left: 1.
So it simplifies to 1.
4 Check
Let's check with a number: x = 30°.
On the calculator, (1 + tan² 30°) × cos² 30° is about 1.0000.
Our answer is 1.
The same!
So it is right.
Let's look back.
First: use sin² + cos² = 1.
Then: write in sin and cos.
Then: cancel.
Last: check.