A3. What is the value of the expression $\frac{\sin 2 \alpha}{1+\tan \alpha}+\left(\cos \alpha+\sin ^{2} \alpha\right)^{2}$ for $\alpha=\frac{\pi}{4}$?
(A) $\frac{\sqrt{2}+5}{4}$
(B) $\frac{\sqrt{2}+6}{4}$
(C) $\frac{5+2 \sqrt{2}}{4}$
(D) $\frac{5-2 \sqrt{2}}{4}$
(E) $5+2 \sqrt{2}$
A3. Insert $\alpha=\frac{\pi}{4} \mathrm{v}$ into the expression and we get $\frac{1}{1+1}+\left(\frac{\sqrt{2}}{2}+\frac{1}{2}\right)^{2}$. Square $\frac{2+2 \sqrt{2}+1}{4}$, add the fractions $\frac{2+2+2 \sqrt{2}+1}{4}=\frac{5+2 \sqrt{2}}{4}$.