Prove that: (sec8θ − 1) / (sec4θ − 1) = tan8θ / tan2θ

(sec8θ − 1) / (sec4θ − 1) = ((1 − cos8θ) cos4θ) / (cos8θ (1 − cos4θ))

= (2sin²4θ × cos4θ) / (cos 8θ × 2 sin²2θ)

= (sin4θ (2 sin4θ × cos4θ)) / (2cos8θ × sin²2θ)

= (sin4θ × sin8θ) / (2cos8θ × sin²2θ)

= (2sin2θ × cos2θ × sin8θ) / (cos8θ × 2sin²2θ)

= tan8θ / tan2θ