$$\begin{gathered} If {(1+x)}^n = C_0+ C_{1}x +C_{2 }{x}^2 + C_{3 }{x}^{3 }+..........+C_r x^{r }+..........+ C_n{x}^{n } \\ then C_o–C_2+C_4–C_6 ...................... = \end{gathered}$$

Put $x=i$ in expansion of ${(1+x)}^n$

$$\begin{gathered} c_0+c_1\left(i\right)+c_2(-1)+c_3(-i)+c_4\left(1\right)+\mathrm{.............}={(1+i)}^n \\ \Rightarrow c_0-c_2+c_4-c_6+\mathrm{........}=Real part of {( 1 + i )}^n \end{gathered}$$
 

= Real part of ${\left(\sqrt{2}(\cos \frac{\pi }{4}+i\sin \frac{\pi }{4})\right)}^n$ $=2^{n/2}.\cos \frac{n\pi }{4}$