x = a cos³θ ----> dx/dθ = −3a cos²θ sinθ

y = a sin³θ ----> dy/dθ = 3a sin²θ cosθ

dy/dx = (dy/dθ) / (dx/dθ)

　　　　= (3a sin²θ cosθ) / (−3a cos²θ sinθ)

　　　　= −sinθ/cosθ

　　　　= −tanθ

Slope = ±1

dy/dx = ±1

−tanθ = ±1

tanθ = ±1

θ = π/4 + kπ/2, for any integer k

When tanθ = 1 (slope = −1)

cosθ = 1/√2 ----> x = a (1/√2)³ = a/(2√2)

sinθ = 1/√2 ----> y = a (1/√2)³ = a/(2√2)

or

cosθ = −1/√2 ----> x = a (−1/√2)³ = −a/(2√2)

sinθ = −1/√2 ----> y = a (−1/√2)³ = −a/(2√2)

When tanθ = −1 (slope = 1)

cosθ = −1/√2 ----> x = a (−1/√2)³ = −a/(2√2)

sinθ = 1/√2 ----> y = a (1/√2)³ = a/(2√2)

or

cosθ = 1/√2 ----> x = a (1/√2)³ = a/(2√2)

sinθ = −1/√2 ----> y = a (−1/√2)³ = −a/(2√2)

Slope = 1 at points (−a/(2√2), a/(2√2)) and (a/(2√2), −a/(2√2))

Slope = −1 at points (a/(2√2), a/(2√2)) and (−a/(2√2), −a/(2√2))

The value of -tan(theta) will be 1 when theta is 45 or 225 or 405 degrees, etc; and -1 when theta is 135 or 315 or 495 degrees, etc.