9. In triangle ABC, right-angled at B, if tan A = 1/√3 find the value of: (i) sin A cos C + cos A sin C (ii) cos A cos C – sin A sin C

Let ΔABC in which ∠B=90° tan A = BC/AB = 1/√3 Let BC = 1k and AB = √3 k, Where k is the positive real number of the problem By Pythagoras theorem in ΔABC we get: AC2=AB2+BC2 AC2=(√3 k)2+(k)2 AC2=3k2+k2 AC2=4k2 AC = 2k Now find the values of cos A, Sin A Sin A = BC/AC = 1/2 Cos A = AB/AC = √3/2 Then find the values of cos C and sin C Sin C = AB/AC = √3/2 Cos C = BC/AC = 1/2 Now, substitute the values in the given problem (i) sin A cos C + cos A sin C = (1/2) ×(1/2 )+ √3/2 ×√3/2 = 1/4 + 3/4 = 1 (ii) cos A cos C – sin A sin C = (√3/2 )(1/2) – (1/2) (√3/2 ) = 0