sin²x + cos²x = 1

Every point on the circle of radius 1 round the origin is exactly 1 from the centre. The point turned through an angle x is (cos(x), sin(x)). Drop a line from it to the x-axis and you have a right triangle: its legs are cos x and sin x long, and its hypotenuse is the radius, 1.

Legs cos x and sin x, hypotenuse 1.

Pythagoras on that triangle gives the identity the rest of this section is built on:

sin(x)² + cos(x)² = 1

sin(x)² is short for (sin(x))², the sine squared. At 60° it reads (√(3)/2)² + (1/2)² = 3/4 + 1/4, which is 1. Because it holds at every angle, it is an identity: you may replace one side by the other wherever that helps. Moving a square across gives two more forms, 1 - sin(x)² = cos(x)² and 1 - cos(x)² = sin(x)².

On the board, 2sin(x)² + 2cos(x)² takes three presses. Factor takes out the 2 and leaves 2(sin(x)² + cos(x)²); the identity turns the bracket into 1; and 2 times 1 is 2.