Question

Prove that cos cube pi by 8 + cos cube cos cube 3 pi by 8 + cos cube 55 by 8 + cos cube 75 by 8 is equal to zero

Answer 1

Answer:

Cos³(π/8) + Cos³(3π/8) + Cos³(5π/8) + Cos³(7π/8)  = 0

Step-by-step explanation:

To be proved

Cos³(π/8) + Cos³(3π/8) + Cos³(5π/8) + Cos³(7π/8)  = 0

LHS

= Cos³(π/8) + Cos³(3π/8) + Cos³(5π/8) + Cos³(7π/8)

= Cos³(π/8) + Cos³(3π/8) + Cos³(π - 3π/8) + Cos³(π - π/8)

as Cos(π - θ) = -Cosθ

= Cos³(π/8) + Cos³(3π/8) + (-Cos³(3π/8)) + (-Cos³(π/8))

= Cos³(π/8) + Cos³(3π/8) - Cos³(3π/8) - Cos³(π/8)

= (Cos³(π/8) - Cos³(π/8)) + (Cos³(3π/8) - Cos³(3π/8))

= 0 + 0

= 0

RHS

QED

Proved

Cos³(π/8) + Cos³(3π/8) + Cos³(5π/8) + Cos³(7π/8)  = 0

Answer 2

Answer:

To be proved

Cos³(π/8) + Cos³(3π/8) + Cos³(5π/8) + Cos³(7π/8) = 0

LHS

= Cos³(π/8) + Cos³(3π/8) + Cos³(5π/8) + Cos³(7π/8)

= Cos³(π/8) + Cos³(3π/8) + Cos³(π - 3π/8) + Cos³(π - π/8)

as Cos(π - θ) = -Cosθ

= Cos³(π/8) + Cos³(3π/8) + (-Cos³(3π/8)) + (-Cos³(π/8))

= Cos³(π/8) + Cos³(3π/8) - Cos³(3π/8) - Cos³(π/8)

= (Cos³(π/8) - Cos³(π/8)) + (Cos³(3π/8) - Cos³(3π/8))

= 0 + 0

= 0

RHS

QED

Proved

Cos³(π/8) + Cos³(3π/8) + Cos³(5π/8) + Cos³(7π/8) = 0