If the function f x -cos 2 x -cos 2-3- x -cos x cos-3- x is constant independent of x - then the value of this constant isA- 1B- 0C-D -

The correct option is C 3/4f(x)=1/2(1+cos 2x)+1/2 ( 1+cos ( 2 π/3+2x ) ) 2 cos x cos ( π/3+x )/2 =1+1/2 [ cos 2x+cos ( 2 π/3+2x ) cos ( 2x+π/3 ) cos ( π/3 ...

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Standard XII

Mathematics

Standard Limits to Remove Indeterminate Form

If the functi...

Question

If the function $f (x) = c o s^{2} x + c o s^{2} (\frac{π}{3} + x) - c o s x c o s (\frac{π}{3} + x)$ is constant (independent of x), then the value of this constant is

\frac{3}{4}

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\frac{4}{3}

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Solution

The correct option is A $\frac{3}{4}$
$f (x) = \frac{1}{2} (1 + c o s 2 x) + \frac{1}{2} (1 + c o s (\frac{2 π}{3} + 2 x)) - \frac{2 c o s x c o s (\frac{π}{3} + x)}{2}$
$= 1 + \frac{1}{2} [c o s 2 x + c o s (\frac{2 π}{3} + 2 x) - c o s (2 x + \frac{π}{3}) - c o s (\frac{π}{3})]$
$= \frac{3}{4} + \frac{1}{2} [c o s 2 x + c o s (2 x + \frac{2 π}{3}) - c o s (2 x + \frac{π}{3})]$
$= \frac{3}{4} + \frac{1}{2} [c o s 2 x - 2 s i n (2 x + \frac{π}{2}) s i n (\frac{π}{6})]$
$= \frac{3}{4} + \frac{1}{2} [c o s 2 x - 2 s i n (\frac{π}{2} + 2 x) . \frac{1}{2}]$
$= \frac{3}{4} + \frac{1}{2} [c o s 2 x - c o s 2 x] = \frac{3}{4}$

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If the function f(x)=cos2x+cos2(π3+x)−cos x cos(π3+x) is constant (independent of x), then the value of this constant is

The correct option is A 34f(x)=12(1+cos 2x)+12(1+cos(2π3+2x))−2 cos x cos(π3+x)2 =1+12[cos 2x+cos(2π3+2x)−cos(2x+π3)−cos(π3)] =34+12[cos 2x+cos(2x+2π3)−cos(2x+π3)] =34+12[cos2x−2sin(2x+π2)sin(π6)] =34+12[cos 2x−2 sin(π2+2x).12] =34+12[cos 2x−cos 2x]=34

If the function $f (x) = c o s^{2} x + c o s^{2} (\frac{π}{3} + x) - c o s x c o s (\frac{π}{3} + x)$ is constant (independent of x), then the value of this constant is