Candace Lawrence Abstract Algebra Task 5 A. Prove that the given function is operation preserving. 2 ix 2 x 2 x cos + isin =e 31 31 31 ( ) ( ) A= 2 x 31 b= 2 ix 31 Now we will create a maclaurin series 1n a2 n +1 Sin a= n=0 ( 2 n+1 ) ! 3 5 7 a a a = a 3 ! + 5! 7 ! ... .. 1n 2 n a Cos a= n=0 2 n! a2 a4 a6 =1- 2 ! + 4 ! 6 ! + ..... e = b n=0 bn n! z 2 z3 1+ z+ + +... . = 2! 3! Now we can plug in our a and b terms. e 2 ix 31 = n=0 2 ix 31 n! 2 3 2 ix 2 ix 31 31 2 ix 1+ + + +... 31 2! 3! ( ) ( ) 2 x 2 2 ix 3 2 ix 4 2 ix 5 31 31 31 31 2 ix 1+ i + +i ... 31 2! 3! 4! 5! ( ) ( ) ( ) ( ) 2 x 2 2 ix 4 2 ix 3 2 ix 5 31 31 31 31 2 ix 1 + ... +i + ... 2! 4! 31 3! 5! ( cos ( ) ( ) )( ( ) ( ) ) ( 231x )+isin ( 231x ) The operation is preserving because addition held on both sides of the equation