Question

eliminate the parameter to find the cartesian equation of the curve for x=tan^2(θ), y=sec(θ), for -π/2<θ<π/2

Answer 1

AAAaannndd the bot gets it wrong yet again!

since sec^2x = 1+tan^2x, we have
y^2 = 1+x

Answer 2

From the given equations, we have:

x = tan^2(θ)

y = sec(θ) = 1/cos(θ)

Multiplying both sides of the second equation by cos(θ), we get:

y cos(θ) = 1

Squaring both sides and using the identity 1 + tan^2(θ) = sec^2(θ), we get:

tan^2(θ) + y^2cos^2(θ) = 1

Substituting x = tan^2(θ) and y = 1/cos(θ), we get:

x + y^2 = 1

This is the Cartesian equation of the curve.