Final answer:
To eliminate the parameter in the parametric equations x = cosh t and y = sinh t, we use the hyperbolic identity cosh^2 t - sinh^2 t = 1, which simplifies to x^2 - y^2 = 1, thus representing a hyperbola.
Step-by-step explanation:
The student is asking to eliminate the parameter in the given parametric equations x = cosh t and y = sinh t. To do this, we can use a property of hyperbolic functions which states that cosh^2 t - sinh^2 t = 1. This relationship is similar to the Pythagorean identity for trigonometric functions but applies to hyperbolic functions.
Since x = cosh t and y = sinh t, squaring both equations and subtracting them will yield x^2 - y^2 = cosh^2 t - sinh^2 t, which simplifies to x^2 - y^2 = 1 based on the hyperbolic identity. So the parameter t is eliminated, and we have an equation in x and y only, representing a hyperbola.
To eliminate the parameter in the given parametric equations, we can express the parameter 't' in terms of 'x' and substitute it into the equation for 'y'. From the equation x = cosh(t), we know that t = cosh^(-1)(x). Substituting this into the equation for y = sinh(t), we have y = sinh(cosh^(-1)(x)). This is the equation that eliminates the parameter.