1. The problem is to evaluate the integral $$\int \cos(\tan x + 1) \sec^2 x \, dx.$$\n\n2. Notice that the integrand contains $\cos(\tan x + 1)$ and $\sec^2 x$. Recall that the derivative of $\tan x$ is $\sec^2 x$, so this suggests a substitution.\n\n3. Let $$u = \tan x + 1.$$ Then, $$\frac{du}{dx} = \sec^2 x,$$ which implies $$du = \sec^2 x \, dx.$$\n\n4. Substitute into the integral: $$\int \cos(u) \, du.$$\n\n5. The integral of $\cos u$ with respect to $u$ is $$\sin u + C.$$\n\n6. Substitute back $u = \tan x + 1$ to get the final answer: $$\sin(\tan x + 1) + C.$$\n\nTherefore, the correct choice is (b) $\sin(\tan x + 1)$.