Prove that :

(1 + tan A. tan B)² + (tan A - tan B)² = sec² A sec² B

If x cos A + y sin A = m and x sin A - y cos A = n, then prove that :

x² + y² = m² + n²

If m = a sec A + b tan A and n = a tan A + b sec A, then prove that :

m² - n² = a² - b²