Find All Complex Solutions sec(x)^2+tan(x)=1

$\sec^{2} (x) + \tan (x) = 1$

Step 1

Subtract $1$ from both sides of the equation.

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

Step 2

Simplify $\sec^{2} (x) + \tan (x) - 1$ .

Move $- 1$ .

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

Apply pythagorean identity.

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

Step 3

Factor $\tan (x)$ out of $\tan^{2} (x) + \tan (x)$ .

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

$\tan (x) \tan (x) + \tan (x) = 0$

Raise $\tan (x)$ to the power of $1$ .

$\tan (x) \tan (x) + \tan (x) = 0$

Factor $\tan (x)$ out of $\tan^{1} (x)$ .

$\tan (x) \tan (x) + \tan (x) \cdot 1 = 0$

Factor $\tan (x)$ out of $\tan (x) \tan (x) + \tan (x) \cdot 1$ .

$\tan (x) (\tan (x) + 1) = 0$

Step 4

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$\tan (x) = 0$

$\tan (x) + 1 = 0$

Step 5

Set $\tan (x)$ equal to $0$ and solve for $x$ .

Set $\tan (x)$ equal to $0$ .

$\tan (x) = 0$

Solve $\tan (x) = 0$ for $x$ .

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

$x = \arctan (0)$

Simplify the right side.

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 of $\tan (x)$ .

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

$\frac{π}{| b |}$

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

$\frac{π}{| 1 |}$

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 $n$

Step 6

Set $\tan (x) + 1$ equal to $0$ and solve for $x$ .

Set $\tan (x) + 1$ equal to $0$ .

$\tan (x) + 1 = 0$

Solve $\tan (x) + 1 = 0$ for $x$ .

Subtract $1$ from both sides of the equation.

$\tan (x) = - 1$

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

$x = \arctan (- 1)$

Simplify the right side.

The exact value of $\arctan (- 1)$ is $- \frac{π}{4}$ .

$x = - \frac{π}{4}$

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

$x = - \frac{π}{4} - π$

Simplify the expression to find the second solution.

Add $2 π$ to $- \frac{π}{4} - π$ .

$x = - \frac{π}{4} - π + 2 π$

The resulting angle of $\frac{3 π}{4}$ is positive and coterminal with $- \frac{π}{4} - π$ .

$x = \frac{3 π}{4}$

Find the period of $\tan (x)$ .

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

$\frac{π}{| b |}$

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

$\frac{π}{| 1 |}$

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 $1$ .

$π$

Add $π$ to every negative angle to get positive angles.

Add $π$ to $- \frac{π}{4}$ to find the positive angle.

$- \frac{π}{4} + π$

To write $π$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$π \cdot \frac{4}{4} - \frac{π}{4}$

Combine fractions.

Combine $π$ and $\frac{4}{4}$ .

$\frac{π \cdot 4}{4} - \frac{π}{4}$

Combine the numerators over the common denominator.

$\frac{π \cdot 4 - π}{4}$

Simplify the numerator.

Move $4$ to the left of $π$ .

$\frac{4 \cdot π - π}{4}$

Subtract $π$ from $4 π$ .

$\frac{3 π}{4}$

List the new angles.

$x = \frac{3 π}{4}$

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

$x = \frac{3 π}{4} + π n, \frac{3 π}{4} + π n$ , for any integer $n$

Step 7

The final solution is all the values that make $\tan (x) (\tan (x) + 1) = 0$ true.

$x = π n, π + π n, \frac{3 π}{4} + π n$ , for any integer $n$

Find All Complex Solutions sec(x)^2+tan(x)=1

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