1. Stating the problem: We need to find the value of $\cos^2 40^\circ + 1$. 2. Recall the Pythagorean identity: $$\sin^2 \theta + \cos^2 \theta = 1$$ which implies $$\cos^2 \theta = 1 - \sin^2 \theta$$ 3. Substitute $\theta = 40^\circ$ into the expression: $$\cos^2 40^\circ + 1 = (1 - \sin^2 40^\circ) + 1 = 2 - \sin^2 40^\circ$$ 4. Therefore, the expression simplifies to: $$2 - \sin^2 40^\circ$$ 5. This matches the second option given. Final answer: $2 - \sin^2 40^\circ$