Prove that cos^2 alpha+sin^2 alpha=1

Answer 1

Prove that: cos² ∝ + sin² ∝ = 1

Detailed Proof:

Draw a right angled triangle with angle ∝ at x-axis. Thus, base = x, height = y, hypotenuses = r

Given/Known: sin ∝ = y/r cos ∝ = x/r x² + y² = r²

Proof: cos² ∝ + sin² ∝ = 1 (start at Left hand side to arrive at Right hand side = 1)

cos² ∝ + sin² ∝ = 1

(cos ∝)² + (sin ∝)² = 1

(x / r)² + (y / r)² = 1

(x² / r²) + (y² / r²) = 1

(x² + y²) / (r²) = 1

(r²) / (r²) = 1

1 = 1 <== Final Proof

Answer 2

Set up a right triangle (1st quadrant is easiest) with hypotenuse = 1, adjacent side x, and opposite side y. So cos α = x/1, sin α = y/1. By the Pythagorean Theorem and substitution, x² + y² = 1² = 1 = cos² α + sin² α.