Proves that sin²θ + cos²θ = 1
The identity that sin²θ + cos²θ = 1 is a fundamental trigonometric identity that holds true for all values of θ.
To prove this identity, we can start with the Pythagorean theorem, which states that in a right-angled triangle, the sum of the squares of the two shorter sides (the legs) is equal to the square of the longest side (the hypotenuse). We can then use this theorem to derive the identity.
Consider a right-angled triangle with angle θ, as shown below:
Let the length of the adjacent side be a and the length of the opposite side be b, and let the length of the hypotenuse be c. Then, using basic trigonometry, we know that:
sin θ = b/c cos θ = a/c
Squaring both of these equations gives:
sin²θ = (b/c)² = b²/c² cos²θ = (a/c)² = a²/c²
Adding these equations together, we get:
sin²θ + cos²θ = b²/c² + a²/c²
Using the Pythagorean theorem, we know that:
a² + b² = c²
Therefore, we can substitute this equation into the above equation to get:
sin²θ + cos²θ = b²/c² + a²/c² = (a² + b²)/c² = c²/c² = 1
This proves that sin²θ + cos²θ = 1, and therefore the identity holds true for all values of θ.