Select your answer Which choice completes the identity? \( \tan ^{2} x= \) \( \begin{array}{l}\sec ^{2} x-1 \\ \csc ^{2} x+1 \\ 2 \sec ^{2} x\end{array} \) \( 1+\cot ^{2} x-1 \)

The identity \( \tan^{2} x = \sec^{2} x - 1 \) is a fundamental trigonometric identity that relates the tangent and secant functions. This relationship comes from the Pythagorean identity \( \sin^{2} x + \cos^{2} x = 1 \). By dividing the entire identity by \( \cos^{2} x \), we derive \( \tan^{2} x = \frac{\sin^{2} x}{\cos^{2} x} \) and \( \sec^{2} x = \frac{1}{\cos^{2} x} \), allowing us to show that \( \tan^{2} x + 1 = \sec^{2} x \). In real-world applications, understanding the relationship between these trigonometric identities can help in various fields such as physics and engineering, particularly in areas involving wave motion and oscillations. For example, when analyzing the angles of a structure or the trajectory of a projectile, recognizing these identities can simplify calculations and lead to more efficient designs or solutions.