Question

Prove that: cos 2a = 2 sin^2b + 4cos (a + b) sin a sin b+ cos 2(a + b)​

Answer 1

Step-by-step explanation:

LHS =2sin2B + 4cos(A+B)sinAsinB + cos2(A+B) .................1

cos2(A+B) = 2cos2(A+B) - 1                                                             [using formula ,cos2a=2cos2a - 1]

putting this in 1

LHS= 2sin2B+4cos(A+B)sinAsinB + 2cos2(A+B) - 1

      =2sin2B - 1 + 2cos(A+B)[cos(A+B) + 2sinAsinB]

      =2sin2B - 1 +2cos(A+B)[cosAcosB -sinAsinB +2sinAsinB]                         [using formula,cosA+B =COSACOSB-SINASINB]

      =2sin2B - 1 + 2cos(A+B)[cos(A-B)]                                                [using formula ,cosACOSB +SINASINB=COSA-B]

      =2sin2B - 1 + 2[cos2A -sin2B]                                                            [COS(A+B)COS(A-B)=COS2A-SIN2B]

      =2cos2A-1

      =cos2A = RHS

hence proved....