Step-by-Step Solution
Key Concept: Solving a first-order linear differential equation using the integrating factor method
Given $y = \sin x + \frac{1}{2}\cosec x$, we differentiate to get $\frac{dy}{dx} = 2\cos x - y\cot x = 2\cos x$. This gives the linear differential equation $\frac{dy}{dx} + y\cot x = 2\cos x$. The integrating factor is $e^{\int \cot x\, dx} = e^{\ln(\sin x)} = \sin x$. Multiplying both sides by $\sin x$ and integrating: the solution is $y\sin x = \int 2\cos x \sin x\, dx + c = \sin^2 x + c$, which matches the given form.
Correct Answer: 2