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Question
Prove the following:
`4 cos x. cos(x + pi/3) . cos (x - pi/3)` = cos 3x
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Solution
`4 cos x. cos(x + pi/3). cos (x - pi/3)`
Cosine Addition Formula: cos (a+b) = cosa cosb − sina sinb
Cosine Subtraction Formula: cos (a−b) = cosa cosb + sina sinb
Here, a = x and b = `pi/3`
L.H.S. = `4 cos x. cos(x + pi/3). cos (x - pi/3)`
= `4 cosx. (cos x cos pi/3 - sin x sin pi/3) (cosx cos pi/3 + sinx sin pi/3)`
= `4 cosx (cosx 1/2 - sinx sqrt3/2) (cosx 1/2 + sinx sqrt3/2)` ...`[∵ sin pi/3 = sqrt3/2, cos pi/3 = 1/2]`
= `4 cos (1/2 cosx - sqrt3/2 sinx) (1/2 cosx + sqrt3/2 sinx)`
= `4 cosx[(1/2 cosx)^2 - (sqrt3/2 sin x)^2]`
= `4 cosx (1/4cos^2x - 3/4 sin^2)`
= `4/4(cos^3x - 3 sin^2.cosx)`
= `cos^3x - 3 cosx.sin^2x`
= cos3x - 3 cosx (1 - cos2x)
= cos3x - 3 cosx + 3 cos3x
= 4 cos3x - 3 cosx
= cos 3x
= R.H.S.
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