Trigonometry Examples

$\cot (x) \sec^{4} (x) = \cot (x) + 2 \tan (x) + \tan^{3} (x)$

Step 1

Move all the expressions to the left side of the equation.

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Step 1.1

Subtract $\cot (x)$ from both sides of the equation.

$\cot (x) \sec^{4} (x) - \cot (x) = 2 \tan (x) + \tan^{3} (x)$

Step 1.2

Subtract $2 \tan (x)$ from both sides of the equation.

$\cot (x) \sec^{4} (x) - \cot (x) - 2 \tan (x) = \tan^{3} (x)$

Step 1.3

Subtract $\tan^{3} (x)$ from both sides of the equation.

$\cot (x) \sec^{4} (x) - \cot (x) - 2 \tan (x) - \tan^{3} (x) = 0$

Step 2

Simplify each term.

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Step 2.1

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

$\frac{\cos (x)}{\sin (x)} \sec^{4} (x) - \cot (x) - 2 \tan (x) - \tan^{3} (x) = 0$

Step 2.2

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

$\frac{\cos (x)}{\sin (x)} {(\frac{1}{\cos (x)})}^{4} - \cot (x) - 2 \tan (x) - \tan^{3} (x) = 0$

Step 2.3

Apply the product rule to $\frac{1}{\cos (x)}$ .

$\frac{\cos (x)}{\sin (x)} \cdot \frac{1^{4}}{\cos^{4} (x)} - \cot (x) - 2 \tan (x) - \tan^{3} (x) = 0$

Step 2.4

One to any power is one.

$\frac{\cos (x)}{\sin (x)} \cdot \frac{1}{\cos^{4} (x)} - \cot (x) - 2 \tan (x) - \tan^{3} (x) = 0$

Step 2.5

Multiply $\frac{\cos (x)}{\sin (x)}$ by $\frac{1}{\cos^{4} (x)}$ .

$\frac{\cos (x)}{\sin (x) \cos^{4} (x)} - \cot (x) - 2 \tan (x) - \tan^{3} (x) = 0$

Step 2.6

Cancel the common factor of $\cos (x)$ and $\cos^{4} (x)$ .

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Step 2.6.1

Multiply by $1$ .

$\frac{\cos (x) \cdot 1}{\sin (x) \cos^{4} (x)} - \cot (x) - 2 \tan (x) - \tan^{3} (x) = 0$

Step 2.6.2

Cancel the common factors.

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Step 2.6.2.1

Factor $\cos (x)$ out of $\sin (x) \cos^{4} (x)$ .

$\frac{\cos (x) \cdot 1}{\cos (x) (\sin (x) \cos^{3} (x))} - \cot (x) - 2 \tan (x) - \tan^{3} (x) = 0$

Step 2.6.2.2

Cancel the common factor.

$\frac{\cos (x) \cdot 1}{\cos (x) (\sin (x) \cos^{3} (x))} - \cot (x) - 2 \tan (x) - \tan^{3} (x) = 0$

Step 2.6.2.3

Rewrite the expression.

$\frac{1}{\sin (x) \cos^{3} (x)} - \cot (x) - 2 \tan (x) - \tan^{3} (x) = 0$

Step 2.7

Separate fractions.

$\frac{1}{\cos^{3} (x)} \cdot \frac{1}{\sin (x)} - \cot (x) - 2 \tan (x) - \tan^{3} (x) = 0$

Step 2.8

Convert from $\frac{1}{\sin (x)}$ to $\csc (x)$ .

$\frac{1}{\cos^{3} (x)} \csc (x) - \cot (x) - 2 \tan (x) - \tan^{3} (x) = 0$

Step 2.9

Combine $\frac{1}{\cos^{3} (x)}$ and $\csc (x)$ .

$\frac{\csc (x)}{\cos^{3} (x)} - \cot (x) - 2 \tan (x) - \tan^{3} (x) = 0$

Step 2.10

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

$\frac{\csc (x)}{\cos (x) \cos^{2} (x)} - \cot (x) - 2 \tan (x) - \tan^{3} (x) = 0$

Step 2.11

Separate fractions.

$\frac{1}{\cos^{2} (x)} \cdot \frac{\csc (x)}{\cos (x)} - \cot (x) - 2 \tan (x) - \tan^{3} (x) = 0$

Step 2.12

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

$\frac{1}{\cos^{2} (x)} \cdot \frac{\frac{1}{\sin (x)}}{\cos (x)} - \cot (x) - 2 \tan (x) - \tan^{3} (x) = 0$

Step 2.13

Rewrite $\frac{\frac{1}{\sin (x)}}{\cos (x)}$ as a product.

$\frac{1}{\cos^{2} (x)} (\frac{1}{\sin (x)} \cdot \frac{1}{\cos (x)}) - \cot (x) - 2 \tan (x) - \tan^{3} (x) = 0$

Step 2.14

Simplify.

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Step 2.14.1

Convert from $\frac{1}{\sin (x)}$ to $\csc (x)$ .

$\frac{1}{\cos^{2} (x)} (\csc (x) \frac{1}{\cos (x)}) - \cot (x) - 2 \tan (x) - \tan^{3} (x) = 0$

Step 2.14.2

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

$\frac{1}{\cos^{2} (x)} (\csc (x) \sec (x)) - \cot (x) - 2 \tan (x) - \tan^{3} (x) = 0$

Step 2.15

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

$\frac{1^{2}}{\cos^{2} (x)} (\csc (x) \sec (x)) - \cot (x) - 2 \tan (x) - \tan^{3} (x) = 0$

Step 2.16

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

${(\frac{1}{\cos (x)})}^{2} (\csc (x) \sec (x)) - \cot (x) - 2 \tan (x) - \tan^{3} (x) = 0$

Step 2.17

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

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

Step 2.18

Multiply $\sec^{2} (x)$ by $\sec (x)$ by adding the exponents.

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Step 2.18.1

Move $\sec (x)$ .

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

Step 2.18.2

Multiply $\sec (x)$ by $\sec^{2} (x)$ .

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Step 2.18.2.1

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

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

Step 2.18.2.2

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

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

Step 2.18.3

Add $1$ and $2$ .

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

Step 3

Set the argument in $\sec (x)$ equal to $\frac{π}{2} + π n$ to find where the expression is undefined.

$x = \frac{π}{2} + π n$ , for any integer $n$

Step 4

Set the argument in $\csc (x)$ equal to $π n$ to find where the expression is undefined.

$x = π n$ , for any integer $n$

Step 5

The equation is undefined where the denominator equals $0$ , the argument of a square root is less than $0$ , or the argument of a logarithm is less than or equal to $0$ .

${x | x = \frac{π}{2} + π n, x = π n}$ $n$ , for any integer $n$

Step 6