$\begin{aligned} & b \cos 2 \theta+a \sin 2 \theta=b \times \frac{1-\tan ^2 \theta}{1+\tan ^2 \theta}+a \times \frac{2 \tan \theta}{1+\tan ^2 \theta} \\ & =b \times \frac{1-\frac{a^2}{b^2}}{1+\frac{a^2}{b^2}}+a \times \frac{2 \times \frac{a}{b}}{1+\frac{a^2}{b^2}} \\ & =\frac{b\left(b^2-a^2\right)}{b^2+a^2}+\frac{2 a^2 b}{b^2+a^2} \\ & =\frac{b\left(b^2+a^2\right)}{b^2+a^2}=b\end{aligned}$