Proof:

To prove that sin²θ + cos²θ = 1, we will use the trigonometric identity:

sin²θ + cos²θ = 1

1. Identity:

The Pythagorean identity is a fundamental trigonometric identity that relates the square of the sine and cosine of an angle. It states that for any angle θ:

sin²θ + cos²θ = 1

This identity is derived from the Pythagorean theorem, which states that in a right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides.

2. Explanation:

To understand why sin²θ + cos²θ equals 1, consider a right triangle with an angle θ. Let's label the sides of the triangle as follows:

- The hypotenuse is h
- The opposite side to angle θ is o
- The adjacent side to angle θ is a

According to the definitions of sine and cosine:

- sinθ = o/h
- cosθ = a/h

Using the Pythagorean theorem, we can express the relationship between the sides of the triangle:

h² = o² + a²

By dividing both sides of the equation by h², we get:

(h²/h²) = (o²/h²) + (a²/h²)

Simplifying further:

1 = (o/h)² + (a/h)²

Substituting sinθ and cosθ back into the equation:

1 = sin²θ + cos²θ

This proves that sin²θ + cos²θ is equal to 1, as stated by the Pythagorean identity.

3. Conclusion:

In conclusion, the equation sin²θ + cos²θ = 1 is a fundamental trigonometric identity known as the Pythagorean identity. It arises from the relationship between the sides of a right triangle and is derived from the Pythagorean theorem. This identity is widely used in various trigonometric calculations and forms the basis for many other trigonometric identities and equations.

Refer maths ncert page no. 190