91Ó°ÊÓ

Trigonometric identities are equations involving trigonometric functions that hold true for all values within their domains. They are incredibly useful, especially in simplifying expressions or connecting different trigonometric functions. One of the fundamental trigonometric identities is \( \sec^2 t = 1 + \tan^2 t \). This particular identity is derived from the Pythagorean identity: \( \sin^2 t + \cos^2 t = 1 \), and it expresses the relationship between the secant and tangent functions.
When dealing with parametric equations, identities like these are crucial because they allow us to link two separate equations into one by eliminating the parameter. By understanding and using trigonometric identities effectively, we can transform complex trigonometric expressions into simpler algebraic forms.

Parametric equations express a set of related quantities as explicit functions of an independent variable, often referred to as the parameter. In this case, the parameter is \(t\), and the equations given are \(x = \tan t\) and \(y = \sec t\).
The goal of eliminating the parameter is to find a direct relationship between \(x\) and \(y\) without involving the parameter \(t\).

• Step 1: Use \(x = \tan t\) and \(y = \sec t\) to connect them through a known trigonometric identity.
• Step 2: Substitute these values into \(\sec^2 t = 1 + \tan^2 t\) to find a relationship between \(x\) and \(y\).
• Step 3: The resulting equation \(y^2 = 1 + x^2\) represents a direct connection between \(x\) and \(y\).
  

Thus, parametric equations provide a powerful way to describe curves and shapes in mathematics by making use of different parameters.

A hyperbola is a type of conic section that appears as two mirror-image curves hovering around the center axis. When we eliminated the parameter \(t\) in the exercise, we ended up with the equation \(y^2 - x^2 = 1\).
This is the standard form of the hyperbola equation, which tells us several things about the curve.

• The equation indicates that the hyperbola is centered at the origin \(0, 0\) and that it opens horizontally.
• This particular form \(y^2 - x^2 = 1\) resembles \(\frac{y^2}{1} - \frac{x^2}{1} = 1\), showing the transverse axis (associated with \(y\)) and the conjugate axis (associated with \(x\)).

Understanding the hyperbola equation is crucial in graphing these curves and analyzing their properties. The process of eliminating the parameter showcases how parametric equations can define complex geometric shapes like hyperbolas through simpler algebraic forms.