Let the function f: (0, π) → ℝ be defined by f(θ) = (sinθ + cosθ) ^...

$f\left(\theta\right) = 1 + \sin 2\theta + {\left(1 - \sin 2\theta\right)}^{2}$

$= \sin^{2}2\theta - \sin 2\theta + 2$

$= {\left(\sin 2\theta - \dfrac{1}{2}\right)}^{2} + \dfrac{7}{4}$

It has minima when $\sin 2\theta = \dfrac{1}{2} = sin\dfrac{\pi}{6}$

$\theta \in \left(0, \pi\right)$

$2\theta = \dfrac{\pi}{6}, \dfrac{5\pi}{6}$

$\theta = \dfrac{\pi}{12}, \dfrac{5\pi}{12}$

${\lambda}_{1} = \dfrac{1}{12}, {\lambda}_{2} = \dfrac{5}{12}$

$sum = \dfrac{1}{2}$.