How would I solve $cos x + cos 2 x = 0$ $cos x + cos 2x = 0$ ? Please show steps.

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How would I solve $cos x + cos 2 x = 0$ $cos x + cos 2x = 0$ ? Please show steps.

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Answer 1 · 2015-11-22T21:31:37

We know that

$cos 2 x = {cos}^{2} x - {sin}^{2} x = {cos}^{2} x - (1 - {cos}^{2} x) = 2 {cos}^{2} x - 1$ $cos2x=cos^2x-sin^2x=cos^2x-(1-cos^2x)=2cos^2x-1$

hence the equation is

$cos x + cos 2 x = cos x + 2 {cos}^{2} x - 1$ $cosx+cos2x=cosx+2cos^2x-1$

Hence we have to solve the

$2 {cos}^{2} x + cos x - 1 = 0 \Rightarrow (cos x + 1) \cdot (2 cos x - 1) = 0$ $2cos^2x+cosx-1=0=>(cosx+1)*(2cosx-1)=0$

or

$cos x = - 1 \Rightarrow cos x = cos π \Rightarrow x = 2 \cdot k \cdot π \pm π$ $cosx=-1=>cosx=cospi=>x=2*k*pi+-pi$

and

$2 cos x - 1 = 0 \Rightarrow cos x = \frac{1}{2} \Rightarrow cos x = cos (\frac{π}{3}) \Rightarrow x = 2 \cdot k \cdot π \pm \frac{π}{3}$ $2cosx-1=0=>cosx=1/2=>cosx=cos(pi/3)=>x=2*k*pi+-pi/3$

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How would I solve $cos x + cos 2 x = 0$ $cos x + cos 2x = 0$ ? Please show steps.

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