Proof of sin² + cos² = 1

To prove that sin² + cos² = 1, we will use the Pythagorean identity which states that sin²θ + cos²θ = 1 for any angle θ.

Let's start by considering a right-angled triangle with sides a, b and hypotenuse c, as shown below:

     /|
    / |
   /  |
a /___| c
    b

By definition, we know that:

• sinθ = a / c
• cosθ = b / c

Using these definitions, we can write:

sin²θ + cos²θ = (a/c)² + (b/c)²

Now, we can use the Pythagorean theorem to substitute c² with a² + b²:

sin²θ + cos²θ = (a/c)² + (b/c)²

             = a²/c² + b²/c²

             = (a² + b²) / c²

             = (a² + b²) / (a² + b²)

             = 1

Therefore, we have proved that sin² + cos² = 1 for any angle θ.