# Trig question

I'm on mobile so I can't use latex..

sin^2 theta - cos^2 theta is not the same thing as one?

If I had a problem say... 1 + cos^2 theta would that equal to sin^2 theta? Or -sin^2 theta? Thanks

eumyang
Homework Helper
I'm on mobile so I can't use latex..

sin^2 theta - cos^2 theta is not the same thing as one?[/tex]
No, it isn't.
$$\sin^2 \theta + \cos^2 \theta = 1$$

If I had a problem say... 1 + cos^2 theta would that equal to sin^2 theta? Or -sin^2 theta? Thanks
Neither.
$$1 + \cos^2 \theta = 2 - \sin^2 \theta$$

Do you see why?

I don't understand how there is a two there.

gb7nash
Homework Helper
I don't understand how there is a two there.

Do you understand why $$\sin^2 \theta + \cos^2 \theta = 1$$ is true?

Once you understand that identity, what do you have to do to the equation to get:

$$1 + \cos^2 \theta = 2 - \sin^2 \theta$$

?

$$1 + \cos^2 \theta = 2 - \sin^2 \theta$$

Think of it this way. If you start from the fundamental pythag. identity:
sin²x + cos²x = 1

You can subtract the sine over so you get:

cos²x = 1-sin²x

Now, 2-sin²x is really the same as
1+1-sin²x

isn't it?

Can you see where I am going with this?

$$1 + \cos^2 \theta = 2 - \sin^2 \theta$$

Think of it this way. If you start from the fundamental pythag. identity:
sin²x + cos²x = 1

You can subtract the sine over so you get:

cos²x = 1-sin²x

Now, 2-sin²x is really the same as
1+1-sin²x

isn't it?

Can you see where I am going with this?

I understand how we move the sine to the right. That would equal to 1-sin²x. But where did the 1 come from that made it 1+1 = 2? All I see is 1-sin²x. So -sin²x can also have a 1 infront of it? Sorry for the frustration.

So If I do the same thing for cos²x, then
sin²x = 1-cos²x
sin²x = 1+1-cos²x
sin²x = 2-cos²x ?

vela
Staff Emeritus
Homework Helper
You can't add 1 to only one side of the equation. You have to do the same thing to both sides of the equation.

Oh, I thought that you were trying to verify that identity, but I see that someone else posted it. What exactly is the problem that you are working on, can you post it in it's entirety?

A great deal of times, when a trig identity has something that would use the pyth. identity, the multiplication of conjugates is needed to get it into the right form.

eumyang
Homework Helper
It didn't sound like a HW question, so I just posted what I thought it would simplify to, in order for the OP to see that the Pythagorean identity can be written in different ways.

I understand how we move the sine to the right. That would equal to 1-sin²x. But where did the 1 come from that made it 1+1 = 2? All I see is 1-sin²x. So -sin²x can also have a 1 infront of it? Sorry for the frustration.
The 1 came from the expression that you gave in the OP!
1 + cos2 θ

All we are doing is replacing the cos2 θ with 1 - sin2 θ:
1 + 1 - sin2 θ = 2 - sin2 θ