Trigonometry Examples

$\sec^{2} (x) + 4 \tan^{2} (x) = 1$

Step 1

Replace the $\sec^{2} (x)$ with $1 + \tan^{2} (x)$ based on the $\tan^{2} (x) + 1 = \sec^{2} (x)$ identity.

$(1 + \tan^{2} (x)) + 4 \tan^{2} (x) = 1$

Step 2

Add $\tan^{2} (x)$ and $4 \tan^{2} (x)$ .

$1 + 5 \tan^{2} (x) = 1$

Step 3

Reorder the polynomial.

$5 \tan^{2} (x) + 1 = 1$

Step 4

Move all terms not containing $\tan (x)$ to the right side of the equation.

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Step 4.1

Subtract $1$ from both sides of the equation.

$5 \tan^{2} (x) = 1 - 1$

Step 4.2

Subtract $1$ from $1$ .

$5 \tan^{2} (x) = 0$

Step 5

Divide each term in $5 \tan^{2} (x) = 0$ by $5$ and simplify.

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Step 5.1

Divide each term in $5 \tan^{2} (x) = 0$ by $5$ .

$\frac{5 \tan^{2} (x)}{5} = \frac{0}{5}$

Step 5.2

Simplify the left side.

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Step 5.2.1

Cancel the common factor of $5$ .

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Step 5.2.1.1

Cancel the common factor.

$\frac{5 \tan^{2} (x)}{5} = \frac{0}{5}$

Step 5.2.1.2

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

$\tan^{2} (x) = \frac{0}{5}$

Step 5.3

Simplify the right side.

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Step 5.3.1

Divide $0$ by $5$ .

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

Step 6

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

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

Step 7

Simplify $\pm \sqrt{0}$ .

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Step 7.1

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

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

Step 7.2

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

$\tan (x) = \pm 0$

Step 7.3

Plus or minus $0$ is $0$ .

$\tan (x) = 0$

Step 8

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

$x = \arctan (0)$

Step 9

Simplify the right side.

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Step 9.1

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

$x = 0$

Step 10

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.

$x = π + 0$

Step 11

Add $π$ and $0$ .

$x = π$

Step 12

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

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Step 12.1

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

$\frac{π}{| b |}$

Step 12.2

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

$\frac{π}{| 1 |}$

Step 12.3

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

$\frac{π}{1}$

Step 12.4

Divide $π$ by $1$ .

$π$

Step 13

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 14

Consolidate the answers.

$x = π n$ , for any integer $n$