Prove that $$3 \sin \frac{\pi}{6} \sec \frac{\pi}{3}-4 \sin \frac{5 \pi}{6} \cot \frac{\pi}{4}=1$$

Step 1: Evaluate individual trigonometric values

First, we need to find the values of each trigonometric function in the given expression. We know that $\frac{\pi}{6}$ radians is equal to $30^\circ$, $\frac{\pi}{3}$ radians is equal to $60^\circ$, and $\frac{\pi}{4}$ radians is equal to $45^\circ$. For $\sin \frac{5\pi}{6}$, we use the identity $\sin(\pi - x) = \sin x$.

Step 2: Substitute the values into the expression

Now, substitute the calculated trigonometric values back into the left-hand side of the given equation. This will allow us to simplify the expression.

Step 3: Simplify the expression

Perform the multiplications in each term. In the first term, $3 \times \frac{1}{2} \times 2$ simplifies to $3 \times 1 = 3$. In the second term, $4 \times \frac{1}{2} \times 1$ simplifies to $2 \times 1 = 2$.

Step 4: Calculate the final result

Finally, subtract the second term from the first term to get the final result. The result is $3 - 2 = 1$. Since this matches the right-hand side of the original equation, the identity is proven.

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