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Let X = x/a and Y = y/b

Then we have :

X cos θ + Y sinθ = 1 --- (1) X Sin θ - Y cosθ = 1 --- (2)

(1) * Cos θ + (2) * Sinθ => X (cos² θ + Sin² θ) = Cos θ + Sinθ so X = cos θ + Sin θ

now (1) * Sinθ - (2) * Cosθ => Y(Sin²θ + Cos²θ) = Sinθ - Cosθ So Y = Sinθ - Cosθ