Question

Find the value of cos^2a + cos^2(a + 2pi /3) + cos^2(a - 2pi/3)

Answer 1

LHS=cos2(π7)+cos2(2π7)+cos2(4π7)
using formula cos2θ=12(1+cos2θ)

=12(1+cos(2π7))+12(1+cos(4π7))+12(1+cos(8π7))
=32+12(cos(2π7)+cos(4π7)+cos(8π7))

=32+14sin(π7)(2sin(π7)cos(2π7)+2sin(π7)cos(4π7)+2sin(π7)cos(8π7))

=32+14sin(π7)(sin(3π7)−sin(π7)+sin(5π7)−sin(3π7)+sin(9π7)−sin(7π7))

=32+14sin(π7)(−sin(π7)+sin(π−2π7)+sin(π+2π7)−0)

=32+14sin(π7)(−sin(π7)+sin(2π7)−sin(2π7))

=32+14sin(π7)(−sin(π7))

=32−14=6−14=54