Step 1: Understanding the Concept:
The given problem is a trigonometric equation involving a product of sine and cosine terms.
In trigonometry, equations containing products are often difficult to solve directly.
The primary objective is to transform the product into a single trigonometric function of a multiple angle.
By doing so, we can use the standard general solution formulas for sine, cosine, or tangent.
The fundamental identity to consider here is the double-angle identity for sine, which relates \( \sin \theta \cos \theta \) to \( \sin 2\theta \).
This transformation simplifies the equation from a quadratic-like product into a linear trigonometric form.
Once the equation is in the form \( \sin \theta = k \), we look for the principal value and then apply the general solution.
Step 2: Key Formula or Approach:
1. Double Angle Identity: \[ 2 \sin x \cos x = \sin 2x \]
2. Standard General Solution: If \( \sin \theta = \sin \alpha \), then the general solution is given by: \[ \theta = n\pi + (-1)^n \alpha \quad \text{where } n \in \mathbb{Z} \]
Step 3: Detailed Explanation:
The provided equation is \( \sin x \cos x = \frac{1}{4} \).
To utilize the double-angle identity, we require a coefficient of 2.
Multiplying both sides of the equation by 2, we obtain:
\[ 2 \sin x \cos x = 2 \times \frac{1}{4} \]
Using the identity \( 2 \sin x \cos x = \sin 2x \), the left-hand side becomes:
\[ \sin 2x = \frac{1}{2} \]
Now, we must identify an angle \( \alpha \) (the principal solution) such that \( \sin \alpha = \frac{1}{2} \).
From the standard trigonometric table, we know that \( \sin \frac{\pi}{6} = \frac{1}{2} \).
Thus, the equation can be rewritten as:
\[ \sin 2x = \sin \frac{\pi}{6} \]
By applying the general solution formula for the sine function, where our argument \( \theta \) is \( 2x \) and our principal angle \( \alpha \) is \( \frac{\pi}{6} \), we get:
\[ 2x = n\pi + (-1)^n \frac{\pi}{6} \]
To find the solution for \( x \), we must isolate it by dividing the entire right-hand side by 2.
It is crucial to divide both the \( n\pi \) term and the principal value term:
\[ x = \frac{n\pi}{2} + (-1)^n \frac{\pi}{6 \times 2} \]
\[ x = \frac{n\pi}{2} + (-1)^n \frac{\pi}{12} \]
Comparing this result with the given options, it matches perfectly with option (A).
This solution provides all possible angles \( x \) that satisfy the original equation for any integer \( n \).
Step 4: Final Answer:
The general solution for the given trigonometric equation is \( x = \frac{n\pi}{2} + (-1)^n \frac{\pi}{12} \).