Let x = secθ − cosθ and y = sec⁴θ − cos⁴θ. What is (dy/dx)² equal to?

We have x = secθ − cosθ, so x² = sec²θ + cos²θ − 2. Also y = sec⁴θ − cos⁴θ = (sec²θ − cos²θ)(sec²θ + cos²θ). Note (sec²θ − cos²θ)² = (sec²θ + cos²θ)² − 4 and x² + 4 = sec²θ + cos²θ + 2 effectively gives sec²θ + cos²θ = x² + 2. Differentiating, dx/dθ = secθ tanθ + sinθ and dy/dθ = 4sec³θ·secθ tanθ + 4cos³θ sinθ = 4(sec⁴θ tanθ secθ·... ). Working it out: dy/dx = 4·y'/x' yields (dy/dx)² = 16(y² + 4)/(x² + 4) after simplification using y² + 4 = (sec⁴θ + cos⁴θ)² and x² + 4 = (secθ + cosθ)².