Trigonometry Examples

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

Step 1

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

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

Step 2

Apply the distributive property.

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

Step 3

Rewrite $- 1 \cot (x)$ as $- \cot (x)$ .

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

Step 4

Simplify the left side.

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Step 4.1

Simplify $\sec^{2} (x) \cot (x) - \cot (x)$ .

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Step 4.1.1

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

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Step 4.1.1.1

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

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

Step 4.1.1.2

Factor $\cot (x)$ out of $- \cot (x)$ .

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

Step 4.1.1.3

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

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

Step 4.1.2

Apply pythagorean identity.

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

Step 4.1.3

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

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

Step 4.1.4

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

$\frac{\cos (x)}{\sin (x)} {(\frac{\sin (x)}{\cos (x)})}^{2} = 3$

Step 4.1.5

Apply the product rule to $\frac{\sin (x)}{\cos (x)}$ .

$\frac{\cos (x)}{\sin (x)} \cdot \frac{\sin^{2} (x)}{\cos^{2} (x)} = 3$

Step 4.1.6

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

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Step 4.1.6.1

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

$\frac{\cos (x)}{\sin (x)} \cdot \frac{\sin^{2} (x)}{\cos (x) \cos (x)} = 3$

Step 4.1.6.2

Cancel the common factor.

$\frac{\cos (x)}{\sin (x)} \cdot \frac{\sin^{2} (x)}{\cos (x) \cos (x)} = 3$

Step 4.1.6.3

Rewrite the expression.

$\frac{1}{\sin (x)} \cdot \frac{\sin^{2} (x)}{\cos (x)} = 3$

Step 4.1.7

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

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Step 4.1.7.1

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

$\frac{1}{\sin (x)} \cdot \frac{\sin (x) \sin (x)}{\cos (x)} = 3$

Step 4.1.7.2

Cancel the common factor.

$\frac{1}{\sin (x)} \cdot \frac{\sin (x) \sin (x)}{\cos (x)} = 3$

Step 4.1.7.3

Rewrite the expression.

$\frac{\sin (x)}{\cos (x)} = 3$

Step 4.1.8

Convert from $\frac{\sin (x)}{\cos (x)}$ to $\tan (x)$ .

$\tan (x) = 3$

Step 5

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

$x = \arctan (3)$

Step 6

Simplify the right side.

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Step 6.1

Evaluate $\arctan (3)$ .

$x = 1.24904577$

Step 7

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.

$x = (3.14159265) + 1.24904577$

Step 8

Solve for $x$ .

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Step 8.1

Remove parentheses.

$x = 3.14159265 + 1.24904577$

Step 8.2

Remove parentheses.

$x = (3.14159265) + 1.24904577$

Step 8.3

Add $3.14159265$ and $1.24904577$ .

$x = 4.39063842$

Step 9

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

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Step 9.1

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

$\frac{π}{| b |}$

Step 9.2

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

$\frac{π}{| 1 |}$

Step 9.3

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

$\frac{π}{1}$

Step 9.4

Divide $π$ by $1$ .

$π$

Step 10

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

$x = 1.24904577 + π n, 4.39063842 + π n$ , for any integer $n$

Step 11

Consolidate $1.24904577 + π n$ and $4.39063842 + π n$ to $1.24904577 + π n$ .

$x = 1.24904577 + π n$ , for any integer $n$