If cosθ/(cosecθ+1) + cosθ/(cosecθ-1) = 2, where 0 < θ < 90°, then what is the value of sin⁴θ + cos⁴θ?

Combine: cosθ[(cosecθ-1+cosecθ+1)]/(cosec²θ-1) = cosθ·2cosecθ/cot²θ = 2cosθ·cosecθ·tan²θ = 2sinθ·... = 2cosθ·(sinθ/cos²θ)·sinθ... Simplifying: 2cosθ·cosecθ/cot²θ = 2cosθ·(1/sinθ)·(sin²θ/cos²θ) = 2sinθ/cosθ = 2tanθ = 2, tanθ=1, θ=45°. sin⁴45+cos⁴45 = 2·(1/4) = 1/2.