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One key to solving this trigonometric identity is employing the sine addition formula. This formula helps us find the sine of a sum of two angles. It is expressed as:

• \( \sin(a + b) = \sin a \cos b + \cos a \sin b \)

This means that the sine of two added angles is equal to the sine of the first angle times the cosine of the second angle plus the cosine of the first angle times the sine of the second angle.
In our case, we need to express \( \sin \left( \frac{\pi}{6} + x \right) \) using this formula, where \( a = \frac{\pi}{6} \) and \( b = x \).
By applying this formula, we can deconstruct the left-hand side of our given identity into its fundamental sine and cosine components.

Knowing the exact values of cosine and sine for frequently occurring angles, such as \( \frac{\pi}{6} \), is crucial when working with trigonometric identities.
Let's break down the values for these specific trigonometric functions:

• \( \sin\left(\frac{\pi}{6}\right) = \frac{1}{2} \)
• \( \cos\left(\frac{\pi}{6}\right) = \frac{\sqrt{3}}{2} \)

These values are derived from the unit circle and are essential basic building blocks when proving identities.
Substituting these values into our transformed equation after applying the sine addition formula simplifies our expression remarkably, allowing us to directly compare it to the right-hand side of the identity we wish to prove.

Once we have rewritten the expression for \( \sin \left( \frac{\pi}{6} + x \right) \) using the sine addition formula and inserted the known sine and cosine values, we're ready to prove the identity.
We simplify our expression to \( \frac{1}{2}\cos x + \frac{\sqrt{3}}{2}\sin x \). Let's compare it to the right-hand side of the given identity: \( \frac{1}{2}(\cos x + \sqrt{3} \sin x) \).
Upon inspection, both expressions are identical, confirming that the given identity holds true. Proving trigonometric identities often involves recognizing expressions on one side of the identity that match those on the other side, using known formulas and values.
This step-by-step process shows how trigonometric formulas like the sine addition formula can be leveraged effectively to prove identities, deepening our understanding of mathematical relationships within trigonometry.