Find and equation equivalent to x^2-y^2=4 in polar coordinates

I’ll give it a shot
x^2-y^2=4
x=rcosθ
y=rsinθ
substitute
r^2cos^2θ-r^2sin^2θ=4
sin^2θ+cos^2θ=1
therefore cos^2θ=1-sin^2θ
substitute
r^2(1-sin^2θ)-r^2sin^2θ=4
r^2-r^2sin^2θ-r^2sin^2θ=4
r^2-2r^2sin^2θ=4
r^2(1-2sin^2θ)=4
r^2=4/(1-2sin^2θ)
take the square root of both sides and separate the numerator and denominator in the square root
r=√4/√(1-2sin^2θ)
r=2/√(1-2sin2θ)

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