Which of the following identities represents tan^2(x)?

The identity representing tan²(x) is derived from the fundamental definitions and relationships within trigonometry. The tangent function is defined as the ratio of the sine function to the cosine function, that is:

tan(x) = sin(x) / cos(x).

When we square both sides of this equation, we arrive at:

tan²(x) = (sin(x) / cos(x))² = sin²(x) / cos²(x).

This identity highlights how tan²(x) is fundamentally based on the square of sine divided by the square of cosine. Therefore, the representation tan²(x) = sin²(x) / cos²(x) is entirely accurate.

Additionally, the other options provided can illustrate different aspects of tangent but do not directly represent tan²(x) in the same manner as this first option does. For instance, sec²(x) - 1 equals tan²(x) based on the Pythagorean identity, but it is not the most direct representation in terms of sine and cosine. Understanding these relationships deepens comprehension of how different trigonometric identities interconnect.