If sin θ – cos θ = 0 then the value of (sin4θ + cos4θ) is

$\sin \theta-\cos \theta=0$

$\Rightarrow \sin \theta=\cos \theta$

$\Rightarrow \frac{\sin \theta}{\cos \theta}=1$

$\Rightarrow \tan \theta=\tan 45^{\circ}$

$\Rightarrow \theta=45^{\circ}$

$\therefore \sin ^{4} \theta+\cos ^{4} \theta$

$=\sin ^{4} 45^{\circ}+\cos ^{4} 45^{\circ}$

$=\left(\frac{1}{\sqrt{2}}\right)^{4}+\left(\frac{1}{\sqrt{2}}\right)^{4}$

$=\frac{1}{4}+\frac{1}{4}$

$=\frac{2}{4}$

$=\frac{1}{2}$

Hence, the correct answer is option (b).