Prove that : cos 5A = 16 cos 5 A – 20 cos 3 A + 5 cos A
Text Solution
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We have cos 5A = cos(3A + 2A) = cos 3A cos 2A – sin 3A sin 2A
= (4 cos 3 A – 3 cos A) (2cos 2 A – 1) – (3sin A – 4sin 3 A)(2sin A cos A)
= (4cos 3 A – 3cosA) (2cos 2 A – 1) – (3 – 4 sin 2 A)(2sin 2 A cos A)
= (4cos 3 A – 3cos A)(2cos 2 A – 1) – {3 – 4(1 – cos 2 A)} {2(1 – cos 2 A) cos A}
= (8cos 5 A – 10cos 3 A + 3cosA) – 2cos A(1 – cos 2 A)(4cos 2 A – 1)
= (8cos 5 A – 10cos 3 A + 3cosA) – 2cos A(5cos 2 A – 4cos 4 A – 1) = 16cos 5 A – 20cos 3 A + 5cosA = RHS
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