Question

Prove that: `cos5A=16cos^5A-20cos^3A+5cosA`

Answer 1

`cos5A = cos(3A+2A) = cos3Acos2A - sin3Asin2A`
`=(4cos^3A-3cosA)(2cos^2A -1) - (3sinA - 4sin^3A)(2sinAcosA)`
`=(8cos^5A-10cos^3A+3cosA)- (3-4sin^2A)(2sin^2AcosA)`
`=(8cos^5A-10cos^3A+3cosA)- (3-4(1-cos^2A))(2(1-cos^2A)cosA)`
`=(8cos^5A-10cos^3A+3cosA)- (3-4+4cos^2A)(2cosA - 2cos^3A)`
`=(8cos^5A-10cos^3A+3cosA)- (-8cos^5A+10cos^3A- 2cosA)`
`=16cos^5A-20cos^3A+5cosA= R.H.S.`