Prove that... cos(a)cos(60-a)cos(60+a)=(cos(3a))/4

2
by ajngamer420

2015-04-24T17:49:04+05:30

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We use the identities

can you derive the 2nd identity and thanks if you did
should be easy with angle sum identity : cos(3x) = cos(2x+x) = cos(2x)cos(x)-sin(2x)sin(x) = (2cos^2(x)-1)cos(x) - 2sin(x)cos(x)sin(x) = ...
2015-04-25T05:58:53+05:30

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We know that standard identity:

2 Cos A Cos B =  Cos (A+B) + Cos (A-B)

here,    A = 60-a  and   B = 60+a

Cos (a)  Cos (60 - a )  Cos (60+a)
= Cos (a) * 1/2 * [ Cos (60-a+60+a) + Cos (60-a- 60-a) ]
= Cos (a) *  1/2 * [ Cos 120  + Cos (-2a)  ]
= 1/2 * Cos (a) * [ -1/2 + Cos (2a) ]
= -1/4 * Cos (a) + 1/2 Cos (a)  Cos (2a)       --- apply the above identity again
= -1/4 * Cos (a)  + 1/2  * 1/2 [ Cos 3a + cos a]
= 1/4 * Cos (3a)

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