Step 1: Plan:
Pick a convenient special case to read off the structure, then confirm with the general expansion.
Step 2: Special check:
Take $\theta=\alpha$. Then $a = \cos0 = 1$ and $b = \sin(\alpha-\beta)$. Let $\varphi = \alpha-\beta$, so $b = \sin\varphi$.
The expression is $\cos^2\varphi + 2\sin\varphi\cdot\sin\varphi + 1 = 2 + \sin^2\varphi$. Test the options with $a=1$, $b^2=\sin^2\varphi$: $2a^2+b^2 = 2+\sin^2\varphi$ matches. $a^2+2b^2 = 1+2\sin^2\varphi$, $a^2-2b^2 = 1 - 2\sin^2\varphi$ and $2a^2-b^2 = 2 - \sin^2\varphi$ do not match.
Step 3: General form:
The full expansion (using $\cos^2 v = 1 - b^2$ and $\sin^2 u = 1 - a^2$) gives $\cos^2(\alpha-\beta)+2ab\sin(\alpha-\beta) = a^2+b^2$. Adding $a^2$ gives $2a^2+b^2$.
Final Answer:
The value is $2a^2+b^2$, option (D).
\[ \boxed{2a^2+b^2} \]