That's the Pythagorean identity right there.

So here's another identity.

You know the identity.

And then the decryption function is also the identity.

So if you know the sine of theta, it's very easy to figure out the cosine of theta, right?

So we have a new, useful trig identity.

What if I just want it in terms of cosines?

Plus the sine I'm just using the sine of a plus b.

So let me clear this out and rewrite this identity.

We proved it using the Pythagorean theorem, and that's actually how we came up with this formula in the first place.

For the second choice, the encryption function is the identity, that gives us M back.

That's the first one, I assume, going into this video we know.

For the second choice, the encryption function is the identity.

Cosine of a times cosine of a minus sine of a, sine of a. This identity was the cosine of a plus b identity.

Now I want to use that identity sine squared theta plus cosine squared theta is equal to one.

This is one I guess identity already. Cosine of 2a is equal to the cosine squared of a minus the sine squared of a.

This is kind of the most fundamental identity. The sine squared of a plus the cosine squared of a is equal to 1.

So we could rewrite this identity as being equal to the cosine squared of a minus the sine squared of a.

Now what if we wanted something in terms of the sine squared of a?

And we know the hypotenuse.

I'm just using this identity over here. This is the exact same thing as the tangent of pi over two minus five theta.

So we've got our first main identity here, that the cosin of theta is equal to the sin of pi over two minus theta.

In algebraic terms the identity means that the norm of product of two octonions equals the product of their norms formula_10.

Let's just play with the ratios and see what else we can other identities we can discover.

And we have another identity. Cosine squared of a, sometimes it's called the power reduction identity right there.

Now if we wanted to solve for sine squared of 2a we could add it to both sides of the equation.

It has a material identity and it's translucent and it's porous, and it allows us for a very different notion of what a skin of a building is.

Now what if I wanted to get an identity that gave me what cosine squared of a is in terms of this?

Well, we could use our handy sine of a plus b identity, which we've already proved, so we can use it now.

Well the most basic trig identity, and this comes out of the unit circle definition of trig functions, is that sine squared x plus cosine squared x is equal to 1.

I can always tell the intelligence of an audience based on how much they appreciate Euler's identity, so y'all have ... y'all have passed that, passed that.

So we now have derived another trigonometric identity that might be in the inside cover of your trig, or actually, your calculus book.

We went through this big convoluted proof with this big convoluted shape, but we got this nice symmetric trig identity out of it.

And if we use this trig identity up here, that is equal to sine of a cosine of a plus the sine of a, cosine of a.

I just used this sine of a plus b identity up here and well, want a and b are both a.

I'm now going to do a proof of a trig identity, which I think is pretty amazing. Although, I think, the proof isn't that obvious.

So the other trig identity is that the cosine of a plus b is equal to the cosine of a you don't mix up the cosines and the sines in this situation. Cosine of a times the sine of b.

And actually, I don't even remember if I've taught it already. I mean, you could view this as a trig identity, although that's almost definitional.

This actually becomes really useful when you actually start doing calculus because you have to solve derivatives and integrals that you might have to know the identity.

The sense of communion with heaven and earth which pervades Millet's The Angelus ... makes Van Gogh's pictures of nightlife in the city seem like visions of hell.

I can convert the cosin to sin using this identity over here.

An analogue of this formula also exists for matrix E again for the same mild price of the correction formula_96 formula_97,In particular (similar to the commutative case) if m n , then formula_98 if m n we return to the identity above.

Namely formula_1 formula_2 formula_3 formula_4 formula_5 formula_6 formula_7 formula_8 formula_9First discovered by Ferdinand Degen (Danish) around 1818, the identity was independently rediscovered by John Thomas Graves (1843) and Arthur Cayley (1845).

And then if you've watched the last two proof videos I made, we also know that the sine of let's say a plus b is equal to the sine of a times the cosine of b.

So sine of sixty cos theta, minus, this is just a standard trig identity So it's a difference, sum and difference identity, minus cosine of sixty, minus cosine of sixty times the sine of theta, times the sine of theta and all this is equal to two sine of theta. two sine of theta. well sine of sixty degrees this is the square root of three over two, square root of three over two. Cosine of sixty degrees is one half, is one half.