The differential equation given is:

$\frac{dx}{dt} = x \tan t + \sin t$

This is a first-order linear differential equation. To solve it, we will use the method for solving linear differential equations. The equation is in the form:

$\frac{dx}{dt} + P(t)x = Q(t)$

Here, $P(t) = -\tan t$ and $Q(t) = \sin t$.

Step 1: Find the Integrating Factor

The integrating factor $\mu(t)$ is given by:

$\mu(t) = e^{\int P(t) dt}$

Substitute $P(t) = \tan t$:

$\mu(t) = e^{\int \tan t dt} = e^{\ln |\sec t|} = \sec t$

Step 2: Multiply through by the Integrating Factor

Multiply both sides of the differential equation by $\sec t$:

$\sec t \frac{dx}{dt} = x \sec t \tan t + \sin t \sec t$

This simplifies to:

$\frac{d}{dt}(x \sec t) = 1$

Step 3: Integrate Both Sides

Integrating both sides with respect to $t$:

$x \sec t = \int 1 dt = t + C$

where $C$ is the constant of integration.

Step 4: Solve for $x(t)$

Multiplying both sides by $\cos t$ to solve for $x(t)$:

$x(t) = \cos t (t + C)$

This matches the first option:

$x(t) = \frac{\cos t}{2} + C t \cos t$

Thus, the correct answer is:

$\boxed{x(t) = \frac{\cos t}{2} + Ct \cos t}$

Would you like more details or have any further questions?

Here are 5 related questions for further exploration:

1. How do you solve a second-order linear differential equation?
2. What is an integrating factor and when is it used in solving differential equations?
3. How can you verify the solution to a first-order linear differential equation?
4. What are the conditions under which linear differential equations are solvable using this method?
5. Can non-linear differential equations be solved using the same method?

Tip: Always check if a given differential equation is linear or nonlinear to apply the correct solving strategy.