Trigonometry Examples

$\sec (x) (\sec (x) - \cos (x)) = \tan^{2} (x)$

Step 1

Simplify the left side.

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Step 1.1

Simplify $\sec (x) (\sec (x) - \cos (x))$ .

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Step 1.1.1

Rewrite $\sec (x)$ in terms of sines and cosines.

$\frac{1}{\cos (x)} (\sec (x) - \cos (x)) = \tan^{2} (x)$

Step 1.1.2

Rewrite $\sec (x)$ in terms of sines and cosines.

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

Step 1.1.3

Apply the distributive property.

$\frac{1}{\cos (x)} \cdot \frac{1}{\cos (x)} + \frac{1}{\cos (x)} (- \cos (x)) = \tan^{2} (x)$

Step 1.1.4

Multiply $\frac{1}{\cos (x)} \cdot \frac{1}{\cos (x)}$ .

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Step 1.1.4.1

Multiply $\frac{1}{\cos (x)}$ by $\frac{1}{\cos (x)}$ .

$\frac{1}{\cos (x) \cos (x)} + \frac{1}{\cos (x)} (- \cos (x)) = \tan^{2} (x)$

Step 1.1.4.2

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

$\frac{1}{\cos^{1} (x) \cos (x)} + \frac{1}{\cos (x)} (- \cos (x)) = \tan^{2} (x)$

Step 1.1.4.3

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

$\frac{1}{\cos^{1} (x) \cos^{1} (x)} + \frac{1}{\cos (x)} (- \cos (x)) = \tan^{2} (x)$

Step 1.1.4.4

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{1}{{\cos (x)}^{1 + 1}} + \frac{1}{\cos (x)} (- \cos (x)) = \tan^{2} (x)$

Step 1.1.4.5

Add $1$ and $1$ .

$\frac{1}{\cos^{2} (x)} + \frac{1}{\cos (x)} (- \cos (x)) = \tan^{2} (x)$

Step 1.1.5

Simplify terms.

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Step 1.1.5.1

Rewrite using the commutative property of multiplication.

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

Step 1.1.5.2

Cancel the common factor of $\cos (x)$ .

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Step 1.1.5.2.1

Move the leading negative in $- \frac{1}{\cos (x)}$ into the numerator.

$\frac{1}{\cos^{2} (x)} + \frac{- 1}{\cos (x)} \cos (x) = \tan^{2} (x)$

Step 1.1.5.2.2

Cancel the common factor.

$\frac{1}{\cos^{2} (x)} + \frac{- 1}{\cos (x)} \cos (x) = \tan^{2} (x)$

Step 1.1.5.2.3

Rewrite the expression.

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

Step 1.1.5.3

Simplify each term.

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Step 1.1.5.3.1

Rewrite $1$ as $1^{2}$ .

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

Step 1.1.5.3.2

Rewrite $\frac{1^{2}}{\cos^{2} (x)}$ as ${(\frac{1}{\cos (x)})}^{2}$ .

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

Step 1.1.5.3.3

Convert from $\frac{1}{\cos (x)}$ to $\sec (x)$ .

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

Step 1.1.6

Apply pythagorean identity.

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

Step 2

Since the exponents are equal, the bases of the exponents on both sides of the equation must be equal.

$| \tan (x) | = | \tan (x) |$

Step 3

Solve for $x$ .

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Step 3.1

Rewrite the absolute value equation as four equations without absolute value bars.

$\tan (x) = \tan (x)$

$\tan (x) = - \tan (x)$

$- \tan (x) = \tan (x)$

$- \tan (x) = - \tan (x)$

Step 3.2

After simplifying, there are only two unique equations to be solved.

$\tan (x) = \tan (x)$

$\tan (x) = - \tan (x)$

Step 3.3

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

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Step 3.3.1

For the two functions to be equal, the arguments of each must be equal.

$x = x$

Step 3.3.2

Move all terms containing $x$ to the left side of the equation.

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Step 3.3.2.1

Subtract $x$ from both sides of the equation.

$x - x = 0$

Step 3.3.2.2

Subtract $x$ from $x$ .

$0 = 0$

Step 3.3.3

Since $0 = 0$ , the equation will always be true.

All real numbers

Step 3.4

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

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Step 3.4.1

Move all terms containing $\tan (x)$ to the left side of the equation.

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Step 3.4.1.1

Add $\tan (x)$ to both sides of the equation.

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

Step 3.4.1.2

Add $\tan (x)$ and $\tan (x)$ .

$2 \tan (x) = 0$

Step 3.4.2

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

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Step 3.4.2.1

Divide each term in $2 \tan (x) = 0$ by $2$ .

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

Step 3.4.2.2

Simplify the left side.

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Step 3.4.2.2.1

Cancel the common factor of $2$ .

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Step 3.4.2.2.1.1

Cancel the common factor.

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

Step 3.4.2.2.1.2

Divide $\tan (x)$ by $1$ .

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

Step 3.4.2.3

Simplify the right side.

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Step 3.4.2.3.1

Divide $0$ by $2$ .

$\tan (x) = 0$

Step 3.4.3

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

$x = \arctan (0)$

Step 3.4.4

Simplify the right side.

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Step 3.4.4.1

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

$x = 0$

Step 3.4.5

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 3.4.6

Add $π$ and $0$ .

$x = π$

Step 3.4.7

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

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Step 3.4.7.1

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

$\frac{π}{| b |}$

Step 3.4.7.2

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

$\frac{π}{| 1 |}$

Step 3.4.7.3

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

$\frac{π}{1}$

Step 3.4.7.4

Divide $π$ by $1$ .

$π$

Step 3.4.8

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 4

Consolidate the answers.

$x = π n$ , for any integer $n$