Keenan Santos

2022-07-11

Finding an identity to solve $\mathrm{cos}t-2\mathrm{cos}\left(2t\right)=0$

toriannucz

Expert

One option is to rewrite the second term using the double angle identity
$\mathrm{cos}2t=2{\mathrm{cos}}^{2}t-1,$
which gives the equation
$\mathrm{cos}t-2\left(2{\mathrm{cos}}^{2}t-1\right)=0.$
Expanding (and canceling a sign for presentation) gives
$4{\mathrm{cos}}^{2}t-\mathrm{cos}t-2=0,$
which is a quadratic in $\mathrm{cos}t$, so it can be analyzed using the Quadratic Formula.

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