Prove the identity:
$\frac{2 \cot x}{\csc ^2 x}=\sin 2 x$
We aim to prove the trigonometric identity \(\frac{2 \cot x}{\csc ^2 x} = \sin 2x\). The domain of this identity consists of all real numbers \(x\) such that the expressions are defined. Specifically, since \(\csc x = \frac{1}{\sin x}\) and \(\cot x = \frac{\cos x}{\sin x}\), the identity is valid for all \(x eq n\pi\) where \(n\) is an integer, because \(\sin x\) appears in the denominators. Our strategy is to transform the left-hand side (LHS) into the right-hand side (RHS) using fundamental trigonometric identities.
We begin with the LHS of the given expression:
\[\text{LHS} = \frac{2 \cot x}{\csc ^2 x}.\]
We express both the cotangent and cosecant functions in terms of sine and cosine:
\[\cot x = \frac{\cos x}{\sin x}\]
\[\csc x = \frac{1}{\sin x} \implies \csc^2 x = \left(\frac{1}{\sin x}\right)^2 = \frac{1}{\sin^2 x}.\]
Substituting these into our expression for the LHS, we get:
\[\text{LHS} = \frac{2 \left(\frac{\cos x}{\sin x}\right)}{\frac{1}{\sin^2 x}}.\]
To simplify the complex fraction, we multiply the numerator by the reciprocal of the denominator:
\[\text{LHS} = 2 \cdot \frac{\cos x}{\sin x} \cdot \sin^2 x.\]
Canceling one factor of \(\sin x\) from the numerator and denominator, we obtain:
\[\text{LHS} = 2 \cdot \cos x \cdot \sin x.\]
This simplifies to:
\[\text{LHS} = 2 \sin x \cos x.\]
Recall the double-angle identity for the sine function, which states:
\[\sin 2x = 2 \sin x \cos x.\]
Comparing this with our simplified LHS, we see that:
\[\text{LHS} = 2 \sin x \cos x = \sin 2x.\]
Since the LHS equals the RHS, the identity is proven.
\[\boxed{\frac{2 \cot x}{\csc ^2 x} = \sin 2x}\]
Trigonometric identities are widely used in physics and engineering to analyze wave motion and oscillating systems. For example, when analyzing the motion of a pendulum or the propagation of sound waves, equations often contain complex trigonometric combinations. Simplifying these using identities like the double-angle formula allows scientists to reduce complex differential equations into more manageable forms, enabling easier calculation of period, frequency, or amplitude in physical systems.