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
AAAaannndd the bot gets it wrong yet again!
since sec^2x = 1+tan^2x, we have
y^2 = 1+x
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.