How do I prove the identity sin²θ + cos²θ = 1 and why is it always true?

This identity comes directly from a right-angled triangle and the Pythagoras theorem. Take a right triangle with hypotenuse h, the side opposite to angle θ as p, and the adjacent side as b. Then sinθ = p/h and cosθ = b/h. By Pythagoras, p² + b² = h². Now sin²θ + cos²θ = p²/h² + b²/h² = (p² + b²)/h². Since p² + b² = h², this becomes h²/h² = 1. So sin²θ + cos²θ = 1 for any angle. Because Pythagoras holds for every right triangle, this identity is always true. From it you can derive the other two: dividing by cos²θ gives 1 + tan²θ = sec²θ, and dividing by sin²θ gives 1 + cot²θ = cosec²θ.