Solve for x 3sec(x)^2-3=0

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Trigonometry

$3 \sec^{2} (x) - 3 = 0$

Simplify the left side.

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Simplify with factoring out.

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Factor $3$ out of $3 \sec^{2} (x)$ .

$3 (\sec^{2} (x)) - 3 = 0$

Factor $3$ out of $- 3$ .

$3 \sec^{2} (x) + 3 \cdot - 1 = 0$

Factor $3$ out of $3 \sec^{2} (x) + 3 \cdot - 1$ .

$3 (\sec^{2} (x) - 1) = 0$

Apply pythagorean identity.

$3 \tan^{2} (x) = 0$

Divide each term by $3$ and simplify.

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Divide each term in $3 \tan^{2} (x) = 0$ by $3$ .

$\frac{3 \tan^{2} (x)}{3} = \frac{0}{3}$

Cancel the common factor of $3$ .

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Cancel the common factor.

$\frac{3 \tan^{2} (x)}{3} = \frac{0}{3}$

Divide $\tan^{2} (x)$ by $1$ .

$\tan^{2} (x) = \frac{0}{3}$

Divide $0$ by $3$ .

$\tan^{2} (x) = 0$

Take the $square$ root of both sides of the $equation$ to eliminate the exponent on the left side.

$\tan (x) = \pm \sqrt{0}$

The complete solution is the result of both the positive and negative portions of the solution.

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Simplify the right side of the equation.

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Rewrite $0$ as $0^{2}$ .

$\tan (x) = \pm \sqrt{0^{2}}$

Pull terms out from under the radical, assuming positive real numbers.

$\tan (x) = \pm 0$

$\pm 0$ is equal to $0$ .

$\tan (x) = 0$

Take the inverse tangent of both sides of the equation to extract $x$ from inside the tangent.

$x = \arctan (0)$

The exact value of $\arctan (0)$ is $0$ .

$x = 0$

The tangent function is positive in the first and third quadrants. To find the second solution, add the reference angle from $π$ to find the solution in the fourth quadrant.

$x = π + 0$

Add $π$ and $0$ .

$x = π$

Find the period.

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The period of the function can be calculated using $\frac{π}{| b |}$ .

$\frac{π}{| b |}$

Replace $b$ with $1$ in the formula for period.

$\frac{π}{| 1 |}$

Solve the equation.

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The absolute value is the distance between a number and zero. The distance between $0$ and $1$ is $1$ .

$\frac{π}{1}$

Divide $π$ by $1$ .

$π$

The period of the $\tan (x)$ function is $π$ so values will repeat every $π$ radians in both directions.

$x = π n, π + π n$ , for any integer $n$

Consolidate the answers.

$x = π n$ , for any integer $n$

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