Starting from the left-hand side (LHS) of the identity
Apply the trigonometric identity: $\cos\left(a\right)-\cos\left(b\right)$$=-2\sin\left(\frac{a-b}{2}\right)\sin\left(\frac{a+b}{2}\right)$, where $a=x-\frac{\pi }{6}$ and $b=x+\frac{\pi }{6}$
Simplify the product $-(x+\frac{\pi }{6})$
Combine fractions with common denominator $6$
Cancel like terms $x$ and $-x$
Divide fractions $\frac{\frac{-2\pi }{6}}{2}$ with Keep, Change, Flip: $\frac{a}{b}\div c=\frac{a}{b}\div\frac{c}{1}=\frac{a}{b}\times\frac{1}{c}=\frac{a}{b\cdot c}$
Cancel the fraction's common factor $2$
Use the odd-even identity $\sin(-\theta)=-\sin(\theta)$
The sine of $\frac{\pi }{6}$ equals $\frac{1}{2}$
Multiply the fraction and term in $2\cdot \left(\frac{1}{2}\right)\sin\left(x\right)$
Since we have reached the expression of our goal, we have proven the identity
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