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 $\sin \theta = \cos \theta$, $\theta = 45^\circ$ or $\dfrac{\pi}{4}$, and $\sin^4 \theta + \cos^4 \theta = \dfrac{1}{2}$.

Answer 1

The Correct Option is B

Step 1: Given condition.
$\sin \theta - \cos \theta = 0 \Rightarrow \sin \theta = \cos \theta$
Step 2: Apply Pythagoras identity.
\[ \sin^2 \theta + \cos^2 \theta = 1 \] Since $\sin \theta = \cos \theta$, we can write: \[ 2 \sin^2 \theta = 1 \Rightarrow \sin^2 \theta = \dfrac{1}{2} \] Step 3: Substitute in the expression.
\[ \sin^4 \theta + \cos^4 \theta = 2(\sin^4 \theta) \] \[ = 2 \left( \dfrac{1}{2} \right)^2 = 2 \times \dfrac{1}{4} = \dfrac{1}{2} \]
Step 4: Conclusion.
Hence, $\sin^4 \theta + \cos^4 \theta = \dfrac{1}{2}$.