Prove the identity: sin^2(x)+cos^2(x) = 1

This is one of the most commonly used A level identities which can be proved using only GCSE maths!

Firstly, take an arbitrary right angle triangle with Hypotenuse h, and angle x between h and the adjacent side. (Diagram recommended)

Label the triangle in terms of h and x using simple SOHCAHTOA:

Hypotenuse = h

Adjacent = hcos(x)

Opposite = hsin(x)

Now, using everyone’s favourite theorem (Pythagorean):

h^2 = h^2cos^2(x)+h^2sin^2(x)

Factoring out h^2 on the right hand side:

h^2 = h^2(cos^2(x)+sin^2(x))

Dividing both sides by h^2 to make it explicit:

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