Question

Prove that cos²(A-B)-sin²(A+B)=cos2Acos2B

Answer 1

see below

Explanation:

Prove : cos²(A-B)-sin²(A+B)=cos2Acos2B

LHS,

cos²(A-B) - sin²(A+B) (expand using double angle identities)

=
`(1 + cos [2(A-B)])/(2) - (1 - cos [2(A+B)])/(2)`

=
`(1)/(2) + (1)/(2) cos[2(A-B)] - (1)/(2) + (1)/(2) cos[2(A+B)]`

=
`(1)/(2) cos[2(A-B)] + (1)/(2) cos[2(A+B)]`

=
`(1)/(2) cos(2A-2B) + (1)/(2) cos(2A+2B)` (expand using sum/difference identities)

=
`(1)/(2) (cos2Acos2B + sin2Asin2B ) + (1)/(2) (cos2Acos2B - sin2Asin2B )`

=
`(1)/(2) cos2Acos2B + (1)/(2)sin2Asin2B + (1)/(2) cos2Acos2B - (1)/(2)sin2Asin2B`

=
`(1)/(2) cos2Acos2B + (1)/(2) cos2Acos2B`

=
`cos2Acos2B` (= RHS , Proven)