The number of solutions of the equation cosx+π3cosπ3-x=14cos22x, x∈[-3π,3π] is:

4cos2⁡π3−sin2⁡x=cos2⁡2x414−sin2⁡x=cos2⁡2x1−4sin2⁡x=cos2⁡2x1−2(1−cos⁡2x)=cos2⁡2xcos⁡2x=t −1+2cos⁡2x=cos2⁡2xt2−2t+1=0(t−1)2=0t=1⇒cos⁡2x=12x=2nπx=nπn=−3,−2,−1,0,1,2,3The number of solutions of the equation cosx+π3cosπ3-x=14cos22x, x∈[-3π,3π] is: 7

The number of solutions of the equation cosx+π3cosπ3-x=14cos22x, x∈[-3π,3π] is:

4cos2⁡π3−sin2⁡x=cos2⁡2x414−sin2⁡x=cos2⁡2x1−4sin2⁡x=cos2⁡2x1−2(1−cos⁡2x)=cos2⁡2xcos⁡2x=t −1+2cos⁡2x=cos2⁡2xt2−2t+1=0(t−1)2=0t=1⇒cos⁡2x=12x=2nπx=nπn=−3,−2,−1,0,1,2,3The number of solutions of the equation cosx+π3cosπ3-x=14cos22x, x∈[-3π,3π] is: 7

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The number of solutions of the equation $\cos (x + \frac{π}{3}) \cos (\frac{π}{3} - x) = \frac{1}{4} \cos^{2} 2 x$ , $x \in [- 3 π, 3 π]$ is:

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answer is D.

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Detailed Solution

$\begin{aligned} 4 (\cos^{2} (\frac{π}{3}) - \sin^{2} x) = \cos^{2} 2 x \\ 4 (\frac{1}{4} - \sin^{2} x) = \cos^{2} 2 x \\ 1 - 4 \sin^{2} x = \cos^{2} 2 x \\ 1 - 2 (1 - \cos 2 x) = \cos^{2} 2 x \end{aligned}$

$\cos 2 x = t \begin{array}{l} - 1 + 2 \cos 2 x = \cos^{2} 2 x \\ t^{2} - 2 t + 1 = 0 \\ (t - 1)^{2} = 0 \\ t = 1 \Rightarrow \cos 2 x = 1 \\ 2 x = 2 nπ \\ x = nπ \\ n = - 3, - 2, - 1, 0, 1, 2, 3 \end{array}$