Why does sin^2(x)+cos^2(x)=1?

We can understand this identity in two different, useful ways.

Firstly, we can use Euler's identities from trigonometry to obtain the desired result using straightforward algebra. We know that:

[1] sin(x) = (exp(ix)-(exp(-ix))/2i

[2] cos(x) = (exp(ix)+exp(-ix))/2

Squaring both sides, and using our knowledge of indices as well as the result i^2=-1 gives:

[3] sin^2(x) = -(exp(2ix)+exp(-2ix)-2)/4

[4] cos^2(x) = (exp(2ix)+exp(-2ix)+2)/4

Adding together [3] and [4] gives:

[5] sin^2(x)+cos^2(x) = (exp(2ix)+exp(-2ix)+2-exp(2ix)-exp(-2ix)+2)/4

Which simplifies to give:

[6] sin^2(x)+cos^2(x) = 4/4 = 1

As required.

Secondly, we can intuitively understand the result by using the most basic trigonometric definitions of sine and cosine that you will have encountered in GCSE and before, and by using Pythagoras' theorem.

Using the whiteboard and the mnemonic SOHCAHTOA, we can see that:

[7] sin(x) = O/H

[8] cos(x) = A/H

Where O, A and H represent the Opposite, Adjacent and Hypotenuse of the triangle relative to the angle, x. Squaring both sides gives:

[9] sin^2(x) = O^2/H^2

[10] cos^2(x) = A^2/H^2

Adding together [9] and [10] gives:

[11] sin^2(x)+cos^2(x) = (O^2+A^2)H^2

From Pythagoras' theorem, we know that O^2+A^2=H^2, and so we find that [11] reduces to:

[12] sin^2(x)+cos^2(x) = H^2/H^2 = 1

As required.