Prove sin^2 x + cos^2 x = 1 Easy

The question

Question: show sin² x + cos² x = 1.

This is true for every angle x.

1 Draw the unit circle

The unit circle has radius 1.

The point P at angle x is (cos x, sin x).

Draw a line from P down to the x-axis.

Across is cos x.

Up is sin x.

The longest side is the radius: 1.

Careful!

sin squared x is the sine of x, then squared.

Not the sine of x squared.

2 Use Pythagoras

It is a right triangle.

Pythagoras: across squared plus up squared is the longest side squared.

Books write (cos x)² as cos² x.

And 1 squared is 1.

Swap the order of adding: sin² x + cos² x = 1.

True for every angle!

3 Check

Let's check with a number: x = 30°.

sin 30° is 0.5.

cos 30° is about 0.8660.

Squared: 0.25 and 0.75.

Together: 1.

Let's look back.

First: draw the unit circle.

Then: use Pythagoras.

Last: check.

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Simplify 1 - sin^2 x Medium

The question

Question: simplify 1 − sin² 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.

Moved around, it says 1 − sin² x = cos² x.

Careful!

The square stays: 1 − sin² x is cos² x, with the square.

So we put cos² x in place of (1 − sin² x).

So it simplifies to cos² x.

2 Check

Let's check with a number: x = 30°.

On the calculator, 1 − sin² 30° is about 0.7500.

And cos² 30° is about 0.7500.

The same!

So it is right.

Let's look back.

First: use sin² + cos² = 1.

Last: check.

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Simplify (1 + tan^2 x) cos^2 x Harder

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.

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