Question

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

• A. 1
• B. 0
• C. 2
• D. -1

Hint:

For trigonometric identities, manipulate the given equation to express \( \sin^2 x \) and \( \sin^4 x \) in terms of \( \cos x \).

Answer 1

The Correct Option is A

Given: \[ \cos x + \cos^2 x = 1 \] Rearranging: \[ \cos^2 x = 1 - \cos x \] Since \( \sin^2 x = 1 - \cos^2 x \), substitute \( \cos^2 x \) from the equation: \[ \sin^2 x = 1 - (1 - \cos x) = \cos x \] Now, \( \sin^4 x = (\sin^2 x)^2 = (\cos x)^2 = \cos^2 x \). Thus: \[ \sin^2 x + \sin^4 x = \cos x + \cos^2 x = 1 \]