UGA Math Placement Practice Exam

To determine the expression that accurately represents cot(2x), it is important to use the double angle identities for the cotangent function. The cotangent of double an angle can be derived from the tangent double angle identity.

The angle doubling formula for tangent states:

\[ \tan(2x) = \frac{2\tan(x)}{1 - \tan^2(x)} \]

Given this formula, the cotangent, which is the reciprocal of tangent, can be represented as:

\[ \cot(2x) = \frac{1}{\tan(2x)} = \frac{1 - \tan^2(x)}{2\tan(x)} \]

This expression is exactly what is provided in the option identified. It effectively utilizes the relationship between cotangent and tangent to express cot(2x) in terms of tan(x).

The components of this derivation underline why this option is the correct choice. The proper manipulation of the identity clearly shows how cot(2x) relates back to tan(x). Thus, it is evident that this representation accurately encapsulates the cotangent of double the angle in terms of the tangent of the original angle.