Question

Prove that:
cos 3A + cos 5A + cos 7A + cos 15A = 4 cos 4A cos 5A cos 6A​

Answer 1

Step-by-step explanation:

cos3A+cos5A+cos7A+cos15A=

2cos(3A+5A)/2×cos(5A-3A)/2

+2cos(15A+7A)/2×cos(15A-7A)/2

=2cos4A.cosA+2cos11Acos4A

=2cos4A(cosA+cos11A)

=2cos4A(2cos12A/2.cos10A/2)

=4cos4A cos6A cos5A (Proved)

Answer 2

2cos [5A+3A/2] cos [5A-3A/2]+2 cos [15A+7A 2] cos [15A-7A/2]

=2cos4AcosA + 2cos11Acos4A

=2cos4A [cosA+cos11A]

=2cos4A [2cos [11A + A/2] Cos[11A - A/2]

=2cos4A2cos5Acos6A

=4cos4Acos5Acos6A.

Hope it helps you mate.