why is sin(x) squared plus cos(x) squared 1?

Thinking of sine and cosine as ratios of side lengths in a right angled triangle, sin(x) = o/h and cos(x) = a/h, so the sin(x)^2 + cos(x)^2 becomes (o^2 + a^2)/h^2. By Pyhtagoras, o^2 + a^2 = h^2, so we get h^2/h^2 = 1.

sin/cos = tan is derived similarly, sin/cos = (o/h)/(a/h) = o/a = tan