= Solution
Using the <Pythagorean trigonometric identity> and the <double-angle formula>,
$$
\sin^4\theta+\cos^4\theta=(\sin^2\theta+\cos^2\theta)^2-2\sin^2\theta\cos^2\theta=1-\frac12\sin^2(2\theta).
$$
The required value $1/2$ therefore occurs when $\sin^2(2\theta)=1$. Within $0\le\theta\le\pi$, the only possibilities are $2\theta=\pi/2$ and $3\pi/2$. Hence
$$
\boxed{\theta=\frac\pi4\quad\text{or}\quad\frac{3\pi}4.}
$$
Equivalently equality requires $\sin^2\theta=\cos^2\theta=1/2$.
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