If sum of all the solution of the equation 8 cos x ( cos ( 6 + x ) · cos ( 6 - x ) - 1 2 ) = 1 in [ 0 ,] is k , then k is equal to

Question

If sum of all the solution of the equation $8 \cos x (\cos (\frac{π}{6} + x) \cdot \cos (\frac{π}{6} - x) - \frac{1}{2}) = 1$ in $[0, π]$ is $k π$ , then $k$ is equal to

Smart Achievers · Accepted Answer

(2)

Key idea Apply the identities

$\cos (x + y) \cos (x - y) = \cos^{2} x - \sin^{2} y$ and $\cos 3 x = 4 \cos^{3} x - 3 \cos x$

We have,

$8 \cos x (\cos (\frac{π}{6} + x) \cos (\frac{π}{6} - x) - \frac{1}{2}) = 1$

$\Rightarrow 8 \cos x (\cos^{2} \frac{π}{6} - \sin^{2} x - \frac{1}{2}) = 1$

$\Rightarrow$ $8 \cos x (\frac{3}{4} - \sin^{2} x - \frac{1}{2}) = 1$

$\Rightarrow 8 \cos x (\frac{3}{4} - \frac{1}{2} - 1 + \cos^{2} x) = 1$

$\Rightarrow$ $8 \cos x (\frac{- 3 + 4 \cos^{2} x}{4}) = 1$

$\Rightarrow$ $2 (4 \cos^{3} x - 3 \cos x) = 1$

$\Rightarrow$ $2 \cos 3 x = 1 \Rightarrow \cos 3 x = \frac{1}{2}$

$\Rightarrow$ $3 x = \frac{π}{3}, \frac{5 π}{3}, \frac{7 π}{3} [0 \leq 3 x \leq 3 π]$

$\Rightarrow$ $x = \frac{π}{9}, \frac{5 π}{9}, \frac{7 π}{9}$

Now, $Sum = \frac{π}{9} + \frac{5 π}{9} + \frac{7 π}{9} = \frac{13 π}{9}$

$\Rightarrow$ $k π = \frac{13 π}{9}$

$∴$ $k = \frac{13}{9}$

Solution

Q.
If sum of all the solution of the equation $8 \cos x (\cos (\frac{π}{6} + x) \cdot \cos (\frac{π}{6} - x) - \frac{1}{2}) = 1$ in $[0, π]$ is $k π$ , then $k$ is equal to

1 $\frac{2}{3}$

2 $\frac{13}{9}$

3 $\frac{8}{9}$

4 $\frac{20}{9}$

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Solution

Q. If sum of all the solution of the equation 8cosx(cos(π6+x)·cos(π6-x)-12)=1 in [0,π] is kπ, then k is equal to

1 23

2 139

3 89