`2sec^2 (x) + tan^2 (x) - 3 = 0` Find all the solutions of the equation in the interval `0,2pi)`.

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Chapter 5, 5.3 - Problem 35 - Precalculus (3rd Edition, Ron Larson).
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mathace | (Level 3) Assistant Educator

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Find all solutions to the equation `2sec^2(x)+tan^2(x)-3=0`    in the interval `[0,2pi).`

`2sec^2(x)+tan^2(x)-3=0`

Use the pythagorean identity `sec^2(x)=1+tan^2(x)`  to substitute in for `sec^2(x).`

`2(1+tan^2(x))+tan^2(x)-3=0`

`2+2tan^2(x)+tan^2(x)-3=0`

`3tan^2(x)-1=0`

`tan^2(x)=1/3`

`tan(x)=+-sqrt(1/3)`

`tan(x)=+-sqrt(3)/3`

`x=pi/6,x=(5pi)/6,x=(7pi)/6,x=(11pi)/6` 

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