Question

Value of $\sin^2\theta + \cos^2\theta$:

• A. 0
• B. 1
• C. 2
• D. -1

Hint:

No matter how complex the angle is (e.g., $\sin^2(5x+3) + \cos^2(5x+3)$), as long as the angles are identical, the result is always 1.

Answer 1

The Correct Option is B

The given question asks for the value of \(\sin^2\theta + \cos^2\theta\). This is a fundamental identity in trigonometry known as the Pythagorean identity.

Formula:

The Pythagorean identity states that for any angle \(\theta\), the following holds true:

\(\sin^2\theta + \cos^2\theta = 1\)

Proof:

1. Consider a right triangle where the hypotenuse is of length 1 (unit circle), the opposite side is \(\sin\theta\), and the adjacent side is \(\cos\theta\).
2. According to the Pythagorean theorem in a right triangle, the sum of the squares of the two legs (sides of the triangle) is equal to the square of the hypotenuse.
3. Thus, \((\sin\theta)^2 + (\cos\theta)^2 = 1^2\).
4. Simplifying, we confirm the identity: \(\sin^2\theta + \cos^2\theta = 1\).

Conclusion:

Therefore, the value of \(\sin^2\theta + \cos^2\theta\) is 1.

Correct Option: 1

This identity is true for all values of \(\theta\), making option '1' the correct choice. Options '0', '2', and '-1' are incorrect as they do not satisfy the trigonometric identity.