The Pythagorean Theorem states that for a right-angled triangle 4ABC, as in figure, a2 =b2 + c2. Use this equation to prove that sin2 θ + cos2 θ = 1

Sure, let's prove that sin²θ + cos²θ = 1 using the Pythagorean theorem.

Step 1: In a right-angled triangle, let's denote the sides as follows:

• The side opposite to angle θ as 'b'
• The side adjacent to angle θ as 'c'
• The hypotenuse as 'a'

Step 2: According to the Pythagorean theorem, we have a² = b² + c².

Step 3: Now, let's express this in terms of trigonometric functions. We know that:

• sinθ = b/a
• cosθ = c/a

Step 4: Squaring both these equations, we get:

• sin²θ = b²/a²
• cos²θ = c²/a²

Step 5: Adding these two equations together, we get: sin²θ + cos²θ = b²/a² + c²/a²

Step 6: Since b²/a² + c²/a² = (b² + c²)/a², we can rewrite the equation as: sin²θ + cos²θ = (b² + c²)/a²

Step 7: But from the Pythagorean theorem, we know that (b² + c²)/a² = 1.

Therefore, we have proved that sin²θ + cos²θ = 1.