\sum_{r=0}^{9}{\left(-1 \right)^r\cos^{10}\frac{r\pi}{10}}
any short solution using complex?
1let \ 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 ?}
4Let 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