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$\begin{array}{*{20}{c}}{\frac{{\sin 0 + \sin 1 + \sin 2 + .......... + \sin n}}{{\cos 0 + \cos 1 + \cos 2 + ......... + \cos n}}}\\\parallel \\{\tan (?)}\end{array}$

L.H.S. $= \frac{{\frac{{\sin \left( {0 + \frac{{n + 1 - 1}}{2}.1} \right)\sin \frac{{(n + 1).1}}{2}}}{{\sin \frac{1}{2}}}}}{{\frac{{\cos \left( {0 + \frac{{n + 1 - 1}}{2}.1} \right)\sin \frac{{(n + 1).1}}{2}}}{{\sin \frac{1}{2}}}}}$

(note that there are n+1 terms in both the numerator and the denominator)

$= \frac{{\sin \frac{n}{2}}}{{\cos \frac{n}{2}}} = \tan \frac{n}{2}$

$\therefore \, ? = \frac{n}{2}$

Sum of the coefficients in the expansion of $(x+y)^n$ ....

If the sum of the coefficients in the expansion of $(x+y)^n$ is 4096, then the greatest coefficient in the expansion is _ _ _ _ . Solution $C_0 + C_1 + C_2 + C_3 + ......................... + C_n =4096$ $\therefore 2^n = 4096 =2^{12}$ $\Rightarrow n = 12$ Greatest coefficient = ${}^{12}{C_6} = 924$