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### $cos^{-1}cos(-5)+$$sin^{-1}sin6$-$tan^{-1}tan12 =?$

$cos^{-1}(cos(-5))+sin^{-1}(sin(6))-tan^{-1}(tan(12))$ is equal to:

(The inverse trigonometric functions take the principal values)

(A) $3\pi - 11$
(B) $3\pi + 1$
(C) $4\pi - 11$
(D) $4\pi - 9$

Solution

The given expression can be rewritten as,

$cos^{-1}cos5+sin^{-1}sin 6-tan^{-1} tan12$

Considering the principal values the given expression can be further rewritten as,

$cos^{-1}cos(2\pi - 5)+sin^{-1}sin (-(2\pi- 6))-tan^{-1} tan(-(4\pi -12))$

Or $(2\pi - 5)+ (-(2\pi- 6))-(-(4\pi-12)) = 4\pi -11$

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