Prove that trignometry identity

Question

a. 2 sin 127 1/2º cos 97 1/2º = 1/2 ( 1 - √2 )

b. 2 cos 127 1/2º  sin 97 1/2º = - 1/2 ( 1 + √2 )

Answer 1

a)

From the trignometric sum rule

2sinAcosB =sin(A+B)+sin(A-B)

2 sin 127 1/2º cos 97 1/2º= sin(127 1/2º+97 1/2º)+sin(127 1/2º-97 1/2º)

                                                            = sin(225º)+sin(30º)

                                                           = sin(180+45º)+sin(30º)

                                                           = -sin(45º)+sin(30º)

                                                           = (-1/√2)+(1/2)

                                                            = (-√2/2)+(1/2)

                                                          = 1/2(1-√2)

Answer 2

b)

From the trignometric sum rule

2cosAsinB =sin(A+B)+sin(A-B)

2 sin 127 1/2º cos 97 1/2º= sin(127 1/2º+97 1/2º)-sin(127 1/2º-97 1/2º)

                                                            = sin(225º)-sin(30º)

                                                           = sin(180+45º)-sin(30º)

                                                           = -sin(45º)-sin(30º)

                                                           = (-1/√2)-(1/2)

                                                            = (-√2/2)-(1/2)

                                                          = -1/2(1+√2)