Problem On Complex Number

[Lost souls]

The problem is:
Show that ((1+cos7Θ)/(1+cosΘ)) = (x⁴-4x²+2)² where x=2cosΘ.

i'm stuck right at the begining. .the expansion for cos7Θ would be
cos⁷Θ - ⁷C₂ cos⁵Θ sin²Θ +⁷C₄ cos³Θ sin⁴Θ - ⁷C₆ cosΘ sin⁶Θ
and then converting the sin²Θ to (1 - cos²Θ) it would be (92 cos⁷Θ - 196 cos⁵Θ +140 cos³Θ - 35cosΘ)
but i cant think of a way that the rhs and lhs will be equal. .is there some substitution that i'm missing or some trick?

 

[HallsofIvy]

The problem is:
Show that ((1+cos7Θ)/(1+cosΘ)) = (x⁴-4x²+2)² where x=2cosΘ.

i'm stuck right at the begining. .the expansion for cos7Θ would be
cos⁷Θ - ⁷C₂ cos⁵Θ sin²Θ +⁷C₄ cos³Θ sin⁴Θ - ⁷C₆ cosΘ sin⁶Θ
and then converting the sin²Θ to (1 - cos²Θ) it would be (92 cos⁷Θ - 196 cos⁵Θ +140 cos³Θ - 35cosΘ)
but i cant think of a way that the rhs and lhs will be equal. .is there some substitution that i'm missing or some trick?

You titled this "Problem on Complex Number". Have you tried using \(\displaystyle cos(7\theta)= \frac{e^{7i\theta}+ e^{-7i\theta}}{2}\), etc.

 

[Lost souls]

You titled this "Problem on Complex Number". Have you tried using \(\displaystyle cos(7\theta)= \frac{e^{7i\theta}+ e^{-7i\theta}}{2}\), etc.

i tried but couldn't get anywhere. .