If `n=12 m(m in N),` prove that `^n C_0-(^n C_2)/((2+sqrt(3))^2)+(^n C_4)/((2+sqrt(3))^4)-(^n C_6)/((2+sqrt(3))^6)+=` ` (-1)^m((2sqrt(2))/(1+sqrt(3)))^ndot`

Explanation

`.(n)C_(0)-(.^(n)C_(2))/((2+sqrt(3))^(2))+(.^(n)C_(4))/((2+sqrt(3))^(4))-(.^(n)C_(6))/((2+sqrt(3))^(6))+"...."`
= Real part of `(1+(i)/(2sqrt(3)))^(n)`
= Real part of `(1+i(2-sqrt(3))^(n)`
= Real part of `(1+ I tan'(pi)/(12))^(n)`
= Real part of `((cos'pi/12+isin'(pi)/(12))^(n))/(cos^(n)'(pi)/(12))`
= Real part of `((cos' (npi)/(12)+isin'(npi)/(12)))/(cos^(n)'(pi)/(12))`
` = (cos'(npi)/(12))/(cos^(n)'(pi)/(12)) = (cos mpi)/(cos^(n)'(pi)/(12))`
` = (-1)^(m)((2sqrt(2))/(1+sqrt(3)))^(n) , [:' cos'(pi)/(12) = (sqrt(3) + 1)/(2sqrt(2))]`