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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.

Solve the differential equation: \( \frac{dx}{dt} = x \tan t + \sin t \).

Learn how to solve the first-order linear differential equation \( \frac{dx}{dt} = x \tan t + \sin t \) using the integrating factor method. This detailed step-by-step guide covers the core concepts of differential equations, integrating factors, and solving techniques, making it suitable for undergraduate calculus students.

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