If ( 18 ) ( 5 18 ) ( 7 18 ) = K , then the value of ( 10K 3 ) is :

Step-by-step solution

Given K = ( 18 ) ( 5 18 ) ( 7 18 ) Using the standard trigonometric identity ( ) ( 3 - ) ( 3 + ) = 1 4 (3 ) with = 18 : 3 - = 6 18 - 18 = 5 18 3 + = 6 18 + 18 = 7 18 Applying the identity, we get: K = 1 4 (3 18 ) = 1 4 ( 6 ) K = 1 4 1 2 = 1 8 We need to find the value of ( 10K 3 ) . Substituting K = 1 8 : ( 10 1 8 3 ) = ( 10 24 ) = ( 5 12 ) Expanding 5 12 as 4 + 6 : ( 5 12 ) = ( 4 + 6 ) ( 5 12 ) = ( 4 ) ( 6 ) + ( 4 ) ( 6 ) ( 5 12 ) = ( 1 2 ) ( 3 2 ) + ( 1 2 ) ( 1 2 ) = 3 + 1 2 2 Answer: 3 + 1 2 2