Prove the identity:
$\frac{2 \cot x}{\csc ^2 x}=\sin 2 x$
To prove the identity \(\frac{2 \cot x}{\csc^2 x} = \sin 2x\), we begin by examining the left-hand side (LHS) of the equation:
\[\text{LHS} = \frac{2 \cot x}{\csc^2 x}\]
Our objective is to simplify this expression using standard trigonometric identities until we arrive at the right-hand side (RHS), which is \(\sin 2x\).
We use the fundamental quotient and reciprocal identities to rewrite the trigonometric functions in terms of \(\sin x\) and \(\cos x\).
Specifically, we know that \(\cot x = \frac{\cos x}{\sin x}\) and \(\csc x = \frac{1}{\sin x}\).
Substituting these into our LHS expression, we get:
\[\text{LHS} = \frac{2 \cdot \frac{\cos x}{\sin x}}{\left(\frac{1}{\sin x}\right)^2} = \frac{2 \cdot \frac{\cos x}{\sin x}}{\frac{1}{\sin^2 x}}\]
Now we perform the algebraic division by multiplying the numerator by the reciprocal of the denominator:
\[\text{LHS} = \left( 2 \cdot \frac{\cos x}{\sin x} \right) \cdot \sin^2 x\]
We can simplify this by canceling one factor of \(\sin x\) in the denominator with one factor of \(\sin x\) in the numerator:
\[\text{LHS} = 2 \cdot \cos x \cdot \sin x = 2 \sin x \cos x\]
Finally, we recall the double-angle identity for the sine function, which states that \(\sin 2x = 2 \sin x \cos x\).
Comparing this to our simplified expression, we see that:
\[\text{LHS} = 2 \sin x \cos x = \sin 2x = \text{RHS}\]
Since the left-hand side matches the right-hand side, the identity is proven. \(\boxed{\frac{2 \cot x}{\csc^2 x} = \sin 2x}\)
Trigonometric identities like this are foundational in fields such as wave mechanics and signal processing. For instance, when engineers analyze sound waves or electrical signals that fluctuate periodically, they use identities to simplify complex equations into manageable forms. Understanding these transformations allows for the easier calculation of wave interference patterns, resonance frequencies, and the design of communication systems.