b) \( \cos ^{2} \theta+2 \sin 2 \theta+3 \sin ^{2} \theta= \)

To solve the equation \( \cos ^{2} \theta + 2 \sin 2 \theta + 3 \sin ^{2} \theta \), we can first remember that \( \sin 2\theta = 2 \sin \theta \cos \theta \). Substituting that in gives us: \[ \cos^{2} \theta + 2 (2 \sin \theta \cos \theta) + 3 \sin^{2} \theta = \cos^{2} \theta + 4 \sin \theta \cos \theta + 3 \sin^{2} \theta. \] Now we can also use the Pythagorean identity, where \( \sin^{2} \theta = 1 - \cos^{2} \theta \). Substituting \( \sin^{2} \theta \) into the equation results in: \[ \cos^{2} \theta + 4 \sin \theta \cos \theta + 3(1 - \cos^{2} \theta) = \cos^{2} \theta + 4 \sin \theta \cos \theta + 3 - 3\cos^{2} \theta. \] Combining like terms gives: \[ -2\cos^{2} \theta + 4\sin \theta \cos \theta + 3. \] This is the simplified expression for \( \cos ^{2} \theta + 2 \sin 2 \theta + 3 \sin ^{2} \theta \).