Question
Prove the identities
1. cos3Ɵ + cos Ɵ = 4cos^Ɵ - 2 cos Ɵ
2. sin (A+B)sin (A-B)=cos2B + sin^2 A-1
1. cos3Ɵ + cos Ɵ = 4cos^Ɵ - 2 cos Ɵ
2. sin (A+B)sin (A-B)=cos2B + sin^2 A-1
Answers
Reiny
hint for #1
cos 3Ø = cos(2Ø + Ø)
= cos2ØcosØ - sin2ØsinØ
now replace cos2Ø = 2cos^-1 Ø - 1
and sin2Ø = 2sinØcosØ
very carefully expand and simplify
2. Just as carefully replace sin(A+b)
with sinAcosB + cosAsinB
and sin(A-B) with sinAcosB - cosAsinB
multiply them and see what comes out
cos 3Ø = cos(2Ø + Ø)
= cos2ØcosØ - sin2ØsinØ
now replace cos2Ø = 2cos^-1 Ø - 1
and sin2Ø = 2sinØcosØ
very carefully expand and simplify
2. Just as carefully replace sin(A+b)
with sinAcosB + cosAsinB
and sin(A-B) with sinAcosB - cosAsinB
multiply them and see what comes out