# Thread: [SOLVED] Proving trig identities

1. ## [SOLVED] Proving trig identities

Prove the following trig identities starting with the left side.

1) sin^2x + cos^2x = 1

2) (cosx - sinx)^2 + (cosx + sinx)^2 = 2

3) cotx + tanx = secx*cscx

Thank you for any help!

2. Originally Posted by live_laugh_luv27
Prove the following trig identities starting with the left side.

1) sin^2x + cos^2x = 1

2) (cosx - sinx)^2 + (cosx + sinx)^2 = 2

3) cotx + tanx = secx*cscx

Thank you for any help!
1. This seems very weird, it's pretty much the basic proof.

$sin(x) = opp/hyp$
$cos(x) = adj/hyp$

Therefore if we square both:

$sin^2(x) = (opp)^2/(hyp)^2$
$cos^2(x) = (adj)^2/(hyp)^2$

$sin^2(x) + cos^2(x) = \frac{(opp)^2 + (adj)^2}{(hyp)^2}$

By the definition of a right angled triangle and using Pythagoras the right side cancels to 1.

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2. Expand to give:

$cos^2(x) - 2sin(x)cos(x) + sin^2(x) + cos^2(x) + 2sin(x)cos(x) + cos^2(x)$

Remember what $cos^2(x)+sin^2(x)$ is equal to

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3. Rewrite in terms of sin and cos:

$\frac{cos(x)}{sin(x)} + \frac{sin(x)}{cos(x)}$

Give them the same denominator by cross multiplying

$\frac{cos^2(x) + sin^2(x)}{sin(x)cos(x)}$

and cancel to give the rhs

3. I know...here is the exact problem.

What is wrong with the following proof?

sin^2x + cos^2x = 1,
sin^2x + cos^2x = sin^2x + (1 - sin^2x),
sin^2x + (1 - sin^2x) = sin^2x - sin^2x + 1,
= 1.

Is there anything wrong with this?

4. Originally Posted by live_laugh_luv27
I know...here is the exact problem.

What is wrong with the following proof?

sin^2x + cos^2x = 1,
sin^2x + cos^2x = sin^2x + (1 - sin^2x),
sin^2x + (1 - sin^2x) = sin^2x - sin^2x + 1,
= 1.

Is there anything wrong with this?
Yeah, you're using the identity you're trying to prove as part of the proof (the bit in red). This shouldn't be done but I can't remember the name for it

5. Originally Posted by e^(i*pi)
Yeah, you're using the identity you're trying to prove as part of the proof (the bit in red). This shouldn't be done but I can't remember the name for it

Oh, ok. So you can't use that part, because then you would be assuming you already proved the identity, which you didn't.