Question

If \( \sin \theta - \cos \theta = 0 \), then the value of \( \sin^4 \theta + \cos^4 \theta \) will be:

• A. \( \dfrac{1}{4} \)
• B. \( \dfrac{1}{2} \)
• C. \( \dfrac{3}{4} \)
• D. 1

Hint:

When \( \sin \theta = \cos \theta \), the value of \( \theta \) is \( 45^\circ \). Use \( \sin^2 \theta + \cos^2 \theta = 1 \) for simplification.

Answer 1

The Correct Option is B

Step 1: Given condition.
\[ \sin \theta - \cos \theta = 0 \Rightarrow \sin \theta = \cos \theta \]
Step 2: Substitute in the required expression.
\[ \sin^4 \theta + \cos^4 \theta = 2 \sin^4 \theta \] Since \( \sin^2 \theta + \cos^2 \theta = 1 \) and \( \sin \theta = \cos \theta \), \[ 2 \sin^2 \theta = 1 \Rightarrow \sin^2 \theta = \frac{1}{2} \]
Step 3: Find value.
\[ \sin^4 \theta + \cos^4 \theta = 2 \left( \frac{1}{2} \right)^2 = 2 \times \frac{1}{4} = \frac{1}{2} \]
Step 4: Final answer.
\[ \boxed{\frac{1}{2}} \]