In a triangle A B C , let A B = 23 , B C = 3 and C A = 4 . Then the value of cot A + cot C cot B is

Step-by-step solution

cot A + cot C cot B = cos A sin A + cos C sin C cos B sin B = cos A sin C + cos C sin A sin A sin C · sin B cos B = sin ( A + C ) sin A sin C · sin B cos B = sin ( π - B ) sin A sin C · sin B cos B = sin 2 B sin A sin C cos B Let, A B = c ,   B C = a ,   C A = b By using sine rule, we get = b 2 a c cos B = b 2 a c a 2 + c 2 - b 2 2 a c = 2 b 2 a 2 + c 2 - b 2 = 32 9 + 23 - 16 = 32 16 = 2