complex

let \ z=\cos\frac{\pi}{10}+i\sin\frac{\pi}{10}\\ z+\frac{1}{z}=2\cos\frac{\pi}{10} \\ \text{we know }z^{10} = -1\\ \sum(-1)^i\left( {z_i+\frac{1}{z_i}}\right)^{10}=\sum\left( {z_i^2+1}\right)^{10}\\ \text{after this it takes around 10 minutes to evaluate , any chota short cut ?}

Let ure expression be some I

I = -(c¹⁰ pi/10 + c¹⁰ 3pi/10 + c¹⁰ 7 pi/10 + c¹⁰ 9 pi/10)

+ (c¹⁰2pi/10 + c¹⁰4pi/10 + c¹⁰ 6pi/10 + c¹⁰ 8pi/10)

+c¹⁰ 0

= 1

Becz cos 7 pi/10 = - cos ( pi - 3pi/10) = - cos 3pi/10

cos 8 pi/10 = - cos (pi - 2pi/10) = - cos 2pi/10

Similarly the other 2

What's My mistake ??

uttara can u please latexify [1]

My solution is wrong

I'll try it again n post