Question

If \( \sin x + \sin^2 x = 1 \), then the value of \( \cos^2 x + \cos^4 x \) is:

• A. 1
• B. -1
• C. 2
• D. 0

Hint:

Use trigonometric identities such as \( \sin^2 x + \cos^2 x = 1 \) to simplify trigonometric expressions and solve equations involving trigonometric functions.

Answer 1

The Correct Option is A

We are given that \( \sin x + \sin^2 x = 1 \). Rearranging this equation: \[ \sin x + \sin^2 x = 1 \quad \Rightarrow \quad \sin x = 1 - \sin^2 x \] Now, using the identity \( \sin^2 x + \cos^2 x = 1 \), we substitute for \( \cos^2 x \): \[ \cos^2 x = 1 - \sin^2 x \] Now, calculate \( \cos^2 x + \cos^4 x \): \[ \cos^2 x + \cos^4 x = 1 - \sin^2 x + (1 - \sin^2 x)^2 \] Simplifying, we find that this expression equals \( 1 \). Therefore, the correct answer is \( 1 \).