31–36. Eliminating the parameter Eliminate the parameter to express the following parametric equations as a single equation in x and y.


x=tan t, y=sec ² t−1 

Verified step by step guidance

1. Start with the given parametric equations: \(x = \tan t\) and \(y = \sec^{2} t - 1\).
2. Recall the Pythagorean identity involving tangent and secant: \(\sec^{2} t = 1 + \tan^{2} t\).
3. Substitute \(\sec^{2} t\) in the expression for \(y\) using the identity: \(y = (1 + \tan^{2} t) - 1\).
4. Simplify the expression for \(y\) to get \(y = \tan^{2} t\).
5. Since \(x = \tan t\), replace \(\tan t\) with \(x\) in the expression for \(y\) to eliminate the parameter \(t\), resulting in \(y = x^{2}\).

Key Concepts

Parametric Equations

Parametric equations express the coordinates of points on a curve as functions of a parameter, often denoted as t. Instead of y being directly related to x, both x and y depend on t, allowing the description of more complex curves.

Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that hold true for all values of the variables. For example, the identity sec² t - tan² t = 1 is essential for relating sec² t and tan t, which helps eliminate the parameter t.

Eliminating the Parameter

Eliminating the parameter involves rewriting parametric equations to form a single equation in x and y by removing the parameter t. This often requires expressing one variable in terms of t and substituting into the other, using identities or algebraic manipulation.