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### Solve${\cos ^2}x + {\cos ^2}2x + {\cos ^2}3x = 1$ &

Select correct option(s). x is odd multiple of ...

(A) $\frac {\pi}{6}$
(B) $\frac {\pi}{4}$
(C) $\frac {\pi}{3}$
(D) $\frac {\pi}{2}$

Solution

We have, ${\cos ^2}x + {\cos ^2}2x = {\sin ^2}3x$

$\Rightarrow ({\cos ^2}x - {\sin ^2}3x) + {\cos ^2}2x = 0$

$\Rightarrow \cos 4x.\cos 2x + {\cos ^2}2x = 0$

$\Rightarrow \cos 2x(\cos 4x + \cos 2x) = 0$

$\Rightarrow \cos 2x.2\cos 3x\cos x = 0$

$\Rightarrow \cos x.\cos 2x.\cos 3x = 0$

If cos x = 0, then x is odd multiple of $\frac {\pi}{2}$.

If cos 2x = 0, then x is odd multiple of $\frac {\pi}{4}$.

If cos 3x = 0, then x is odd multiple of $\frac {\pi}{6}$.

Answer: (A), (B), (D).

### 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 = ${}^{12}{C_6} = 924$