Prove the following identities, where the angles involved are acute angles for which the expressions are defined. (i) (cosec θ − cot θ)² = 1 − cos θ / 1 + cos θ (ii) cos A / 1 + sin A + 1 + sin A / cos A = 2 sec A (iii) tan θ / 1 − cot θ + cot θ / 1 − tan θ = 1 + sec θ cosec θ (iv) 1 + sec A / sec A = sin² A / 1 − cos A (v) cos A / cos A + sin A / 1 = cosec A + cot A, using the identity cosec² A = 1 + cot² A. (vi) 1 + sin A / 1 − sin A = sec A + tan A (vii) sin θ / 2 cos³ θ − cos θ = tan θ (viii) (sin A + cosec A)² + (cos A + sec A)² = 7 + tan² A + cot² A