The maximum value of cos 2((π/3)-x)- cos 2 ((π/3)+x) is

Question

The maximum value of cos 2((π/3)-x)- cos 2 ((π/3)+x) is (A) -(√3/2) (B) (1/2) (C) (√3/2) (D) (3/2)

Tardigrade Admin · Accepted Answer

cos 2((π/3)-x)- cos 2((π/3)+x) = sin ((π/3)-x+(π/3)+x) sin ((π/3)+x-(π/3)+x) = sin (2 π/3) sin 2 x=(√3/2) sin 2 x (since, maximum value of sin 2 x is 1 ) Therefore, its maximum value is (√3/2).

Q. The maximum value of $cos^{2} (\frac{π}{3} - x) - cos^{2}$ $(\frac{π}{3} + x)$ is

$- \frac{3}{2}$

$\frac{1}{2}$

$\frac{3}{2}$

$\frac{3}{2}$

$cos^{2} (\frac{π}{3} - x) - cos^{2} (\frac{π}{3} + x)$
$= sin (\frac{π}{3} - x + \frac{π}{3} + x) sin (\frac{π}{3} + x - \frac{π}{3} + x)$
$= sin \frac{2 π}{3} sin 2 x = \frac{3}{2} sin 2 x$
(since, maximum value of $sin 2 x$ is 1 )
Therefore, its maximum value is $\frac{3}{2}$ .

Q. The maximum value of cos2(3π​−x)−cos2 (3π​+x) is

−23​​

21​

23​​

23​

cos2(3π​−x)−cos2(3π​+x) =sin(3π​−x+3π​+x)sin(3π​+x−3π​+x) =sin32π​sin2x=23​​sin2x (since, maximum value of sin2x is 1 ) Therefore, its maximum value is 23​​.

Q. The maximum value of $cos^{2} (\frac{π}{3} - x) - cos^{2}$ $(\frac{π}{3} + x)$ is

$- \frac{3}{2}$

$\frac{1}{2}$

$\frac{3}{2}$

$\frac{3}{2}$

$cos^{2} (\frac{π}{3} - x) - cos^{2} (\frac{π}{3} + x)$
$= sin (\frac{π}{3} - x + \frac{π}{3} + x) sin (\frac{π}{3} + x - \frac{π}{3} + x)$
$= sin \frac{2 π}{3} sin 2 x = \frac{3}{2} sin 2 x$
(since, maximum value of $sin 2 x$ is 1 )
Therefore, its maximum value is $\frac{3}{2}$ .