Trigonometry Examples

$\frac{- 1 + \cot^{2} (w) + \cos^{2} (w) \tan^{2} (w)}{\csc^{2} (w)} = \cos^{4} (w)$

Step 1

Subtract $\cos^{4} (w)$ from both sides of the equation.

$\frac{- 1 + \cot^{2} (w) + \cos^{2} (w) \tan^{2} (w)}{\csc^{2} (w)} - \cos^{4} (w) = 0$

Step 2

Simplify $\frac{- 1 + \cot^{2} (w) + \cos^{2} (w) \tan^{2} (w)}{\csc^{2} (w)} - \cos^{4} (w)$ .

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Step 2.1

Simplify each term.

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Step 2.1.1

Simplify the numerator.

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Step 2.1.1.1

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

$\frac{- 1 + {(\frac{\cos (w)}{\sin (w)})}^{2} + \cos^{2} (w) \tan^{2} (w)}{\csc^{2} (w)} - \cos^{4} (w) = 0$

Step 2.1.1.2

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

$\frac{- 1 + \frac{\cos^{2} (w)}{\sin^{2} (w)} + \cos^{2} (w) \tan^{2} (w)}{\csc^{2} (w)} - \cos^{4} (w) = 0$

Step 2.1.1.3

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

$\frac{- 1 + \frac{\cos^{2} (w)}{\sin^{2} (w)} + \cos^{2} (w) {(\frac{\sin (w)}{\cos (w)})}^{2}}{\csc^{2} (w)} - \cos^{4} (w) = 0$

Step 2.1.1.4

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

$\frac{- 1 + \frac{\cos^{2} (w)}{\sin^{2} (w)} + \cos^{2} (w) \frac{\sin^{2} (w)}{\cos^{2} (w)}}{\csc^{2} (w)} - \cos^{4} (w) = 0$

Step 2.1.1.5

Cancel the common factor of $\cos^{2} (w)$ .

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Step 2.1.1.5.1

Cancel the common factor.

$\frac{- 1 + \frac{\cos^{2} (w)}{\sin^{2} (w)} + \cos^{2} (w) \frac{\sin^{2} (w)}{\cos^{2} (w)}}{\csc^{2} (w)} - \cos^{4} (w) = 0$

Step 2.1.1.5.2

Rewrite the expression.

$\frac{- 1 + \frac{\cos^{2} (w)}{\sin^{2} (w)} + \sin^{2} (w)}{\csc^{2} (w)} - \cos^{4} (w) = 0$

Step 2.1.1.6

Move $\sin^{2} (w)$ .

$\frac{- 1 + \sin^{2} (w) + \frac{\cos^{2} (w)}{\sin^{2} (w)}}{\csc^{2} (w)} - \cos^{4} (w) = 0$

Step 2.1.1.7

Rewrite $- 1$ as $- 1 (1)$ .

$\frac{- 1 (1) + \sin^{2} (w) + \frac{\cos^{2} (w)}{\sin^{2} (w)}}{\csc^{2} (w)} - \cos^{4} (w) = 0$

Step 2.1.1.8

Factor $- 1$ out of $\sin^{2} (w)$ .

$\frac{- 1 (1) - 1 (- \sin^{2} (w)) + \frac{\cos^{2} (w)}{\sin^{2} (w)}}{\csc^{2} (w)} - \cos^{4} (w) = 0$

Step 2.1.1.9

Factor $- 1$ out of $- 1 (1) - 1 (- \sin^{2} (w))$ .

$\frac{- 1 (1 - \sin^{2} (w)) + \frac{\cos^{2} (w)}{\sin^{2} (w)}}{\csc^{2} (w)} - \cos^{4} (w) = 0$

Step 2.1.1.10

Rewrite $- 1 (1 - \sin^{2} (w))$ as $- (1 - \sin^{2} (w))$ .

$\frac{- (1 - \sin^{2} (w)) + \frac{\cos^{2} (w)}{\sin^{2} (w)}}{\csc^{2} (w)} - \cos^{4} (w) = 0$

Step 2.1.1.11

Apply pythagorean identity.

$\frac{- \cos^{2} (w) + \frac{\cos^{2} (w)}{\sin^{2} (w)}}{\csc^{2} (w)} - \cos^{4} (w) = 0$

Step 2.1.1.12

Factor $\cos^{2} (w)$ out of $- \cos^{2} (w) + \frac{\cos^{2} (w)}{\sin^{2} (w)}$ .

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Step 2.1.1.12.1

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

$\frac{\cos^{2} (w) \cdot - 1 + \frac{\cos^{2} (w)}{\sin^{2} (w)}}{\csc^{2} (w)} - \cos^{4} (w) = 0$

Step 2.1.1.12.2

Factor $\cos^{2} (w)$ out of $\frac{\cos^{2} (w)}{\sin^{2} (w)}$ .

$\frac{\cos^{2} (w) \cdot - 1 + \cos^{2} (w) \frac{1}{\sin^{2} (w)}}{\csc^{2} (w)} - \cos^{4} (w) = 0$

Step 2.1.1.12.3

Factor $\cos^{2} (w)$ out of $\cos^{2} (w) \cdot - 1 + \cos^{2} (w) \frac{1}{\sin^{2} (w)}$ .

$\frac{\cos^{2} (w) (- 1 + \frac{1}{\sin^{2} (w)})}{\csc^{2} (w)} - \cos^{4} (w) = 0$

Step 2.1.1.13

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

$\frac{\cos^{2} (w) (- 1 + \frac{1^{2}}{\sin^{2} (w)})}{\csc^{2} (w)} - \cos^{4} (w) = 0$

Step 2.1.1.14

Rewrite $\frac{1^{2}}{\sin^{2} (w)}$ as ${(\frac{1}{\sin (w)})}^{2}$ .

$\frac{\cos^{2} (w) (- 1 + {(\frac{1}{\sin (w)})}^{2})}{\csc^{2} (w)} - \cos^{4} (w) = 0$

Step 2.1.1.15

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

$\frac{\cos^{2} (w) (- 1^{2} + {(\frac{1}{\sin (w)})}^{2})}{\csc^{2} (w)} - \cos^{4} (w) = 0$

Step 2.1.1.16

Reorder $- 1^{2}$ and ${(\frac{1}{\sin (w)})}^{2}$ .

$\frac{\cos^{2} (w) ({(\frac{1}{\sin (w)})}^{2} - 1^{2})}{\csc^{2} (w)} - \cos^{4} (w) = 0$

Step 2.1.1.17

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = \frac{1}{\sin (w)}$ and $b = 1$ .

$\frac{\cos^{2} (w) (\frac{1}{\sin (w)} + 1) (\frac{1}{\sin (w)} - 1)}{\csc^{2} (w)} - \cos^{4} (w) = 0$

Step 2.1.2

Simplify the denominator.

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Step 2.1.2.1

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

$\frac{\cos^{2} (w) (\frac{1}{\sin (w)} + 1) (\frac{1}{\sin (w)} - 1)}{{(\frac{1}{\sin (w)})}^{2}} - \cos^{4} (w) = 0$

Step 2.1.2.2

Apply the product rule to $\frac{1}{\sin (w)}$ .

$\frac{\cos^{2} (w) (\frac{1}{\sin (w)} + 1) (\frac{1}{\sin (w)} - 1)}{\frac{1^{2}}{\sin^{2} (w)}} - \cos^{4} (w) = 0$

Step 2.1.2.3

One to any power is one.

$\frac{\cos^{2} (w) (\frac{1}{\sin (w)} + 1) (\frac{1}{\sin (w)} - 1)}{\frac{1}{\sin^{2} (w)}} - \cos^{4} (w) = 0$

Step 2.1.3

Multiply the numerator by the reciprocal of the denominator.

$\cos^{2} (w) (\frac{1}{\sin (w)} + 1) (\frac{1}{\sin (w)} - 1) \sin^{2} (w) - \cos^{4} (w) = 0$

Step 2.1.4

Apply the distributive property.

$(\cos^{2} (w) \frac{1}{\sin (w)} + \cos^{2} (w) \cdot 1) (\frac{1}{\sin (w)} - 1) \sin^{2} (w) - \cos^{4} (w) = 0$

Step 2.1.5

Combine $\cos^{2} (w)$ and $\frac{1}{\sin (w)}$ .

$(\frac{\cos^{2} (w)}{\sin (w)} + \cos^{2} (w) \cdot 1) (\frac{1}{\sin (w)} - 1) \sin^{2} (w) - \cos^{4} (w) = 0$

Step 2.1.6

Multiply $\cos^{2} (w)$ by $1$ .

$(\frac{\cos^{2} (w)}{\sin (w)} + \cos^{2} (w)) (\frac{1}{\sin (w)} - 1) \sin^{2} (w) - \cos^{4} (w) = 0$

Step 2.1.7

Expand $(\frac{\cos^{2} (w)}{\sin (w)} + \cos^{2} (w)) (\frac{1}{\sin (w)} - 1)$ using the FOIL Method.

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Step 2.1.7.1

Apply the distributive property.

$(\frac{\cos^{2} (w)}{\sin (w)} (\frac{1}{\sin (w)} - 1) + \cos^{2} (w) (\frac{1}{\sin (w)} - 1)) \sin^{2} (w) - \cos^{4} (w) = 0$

Step 2.1.7.2

Apply the distributive property.

$(\frac{\cos^{2} (w)}{\sin (w)} \cdot \frac{1}{\sin (w)} + \frac{\cos^{2} (w)}{\sin (w)} \cdot - 1 + \cos^{2} (w) (\frac{1}{\sin (w)} - 1)) \sin^{2} (w) - \cos^{4} (w) = 0$

Step 2.1.7.3

Apply the distributive property.

$(\frac{\cos^{2} (w)}{\sin (w)} \cdot \frac{1}{\sin (w)} + \frac{\cos^{2} (w)}{\sin (w)} \cdot - 1 + \cos^{2} (w) \frac{1}{\sin (w)} + \cos^{2} (w) \cdot - 1) \sin^{2} (w) - \cos^{4} (w) = 0$

Step 2.1.8

Simplify and combine like terms.

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Step 2.1.8.1

Simplify each term.

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Step 2.1.8.1.1

Combine.

$(\frac{\cos^{2} (w) \cdot 1}{\sin (w) \sin (w)} + \frac{\cos^{2} (w)}{\sin (w)} \cdot - 1 + \cos^{2} (w) \frac{1}{\sin (w)} + \cos^{2} (w) \cdot - 1) \sin^{2} (w) - \cos^{4} (w) = 0$

Step 2.1.8.1.2

Multiply $\cos^{2} (w)$ by $1$ .

$(\frac{\cos^{2} (w)}{\sin (w) \sin (w)} + \frac{\cos^{2} (w)}{\sin (w)} \cdot - 1 + \cos^{2} (w) \frac{1}{\sin (w)} + \cos^{2} (w) \cdot - 1) \sin^{2} (w) - \cos^{4} (w) = 0$

Step 2.1.8.1.3

Simplify the denominator.

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Step 2.1.8.1.3.1

Raise $\sin (w)$ to the power of $1$ .

$(\frac{\cos^{2} (w)}{\sin^{1} (w) \sin (w)} + \frac{\cos^{2} (w)}{\sin (w)} \cdot - 1 + \cos^{2} (w) \frac{1}{\sin (w)} + \cos^{2} (w) \cdot - 1) \sin^{2} (w) - \cos^{4} (w) = 0$

Step 2.1.8.1.3.2

Raise $\sin (w)$ to the power of $1$ .

$(\frac{\cos^{2} (w)}{\sin^{1} (w) \sin^{1} (w)} + \frac{\cos^{2} (w)}{\sin (w)} \cdot - 1 + \cos^{2} (w) \frac{1}{\sin (w)} + \cos^{2} (w) \cdot - 1) \sin^{2} (w) - \cos^{4} (w) = 0$

Step 2.1.8.1.3.3

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

$(\frac{\cos^{2} (w)}{{\sin (w)}^{1 + 1}} + \frac{\cos^{2} (w)}{\sin (w)} \cdot - 1 + \cos^{2} (w) \frac{1}{\sin (w)} + \cos^{2} (w) \cdot - 1) \sin^{2} (w) - \cos^{4} (w) = 0$

Step 2.1.8.1.3.4

Add $1$ and $1$ .

$(\frac{\cos^{2} (w)}{\sin^{2} (w)} + \frac{\cos^{2} (w)}{\sin (w)} \cdot - 1 + \cos^{2} (w) \frac{1}{\sin (w)} + \cos^{2} (w) \cdot - 1) \sin^{2} (w) - \cos^{4} (w) = 0$

Step 2.1.8.1.4

Combine $\frac{\cos^{2} (w)}{\sin (w)}$ and $- 1$ .

$(\frac{\cos^{2} (w)}{\sin^{2} (w)} + \frac{\cos^{2} (w) \cdot - 1}{\sin (w)} + \cos^{2} (w) \frac{1}{\sin (w)} + \cos^{2} (w) \cdot - 1) \sin^{2} (w) - \cos^{4} (w) = 0$

Step 2.1.8.1.5

Move $- 1$ to the left of $\cos^{2} (w)$ .

$(\frac{\cos^{2} (w)}{\sin^{2} (w)} + \frac{- 1 \cdot \cos^{2} (w)}{\sin (w)} + \cos^{2} (w) \frac{1}{\sin (w)} + \cos^{2} (w) \cdot - 1) \sin^{2} (w) - \cos^{4} (w) = 0$

Step 2.1.8.1.6

Move the negative in front of the fraction.

$(\frac{\cos^{2} (w)}{\sin^{2} (w)} - \frac{\cos^{2} (w)}{\sin (w)} + \cos^{2} (w) \frac{1}{\sin (w)} + \cos^{2} (w) \cdot - 1) \sin^{2} (w) - \cos^{4} (w) = 0$

Step 2.1.8.1.7

Combine $\cos^{2} (w)$ and $\frac{1}{\sin (w)}$ .

$(\frac{\cos^{2} (w)}{\sin^{2} (w)} - \frac{\cos^{2} (w)}{\sin (w)} + \frac{\cos^{2} (w)}{\sin (w)} + \cos^{2} (w) \cdot - 1) \sin^{2} (w) - \cos^{4} (w) = 0$

Step 2.1.8.1.8

Move $- 1$ to the left of $\cos^{2} (w)$ .

$(\frac{\cos^{2} (w)}{\sin^{2} (w)} - \frac{\cos^{2} (w)}{\sin (w)} + \frac{\cos^{2} (w)}{\sin (w)} - 1 \cdot \cos^{2} (w)) \sin^{2} (w) - \cos^{4} (w) = 0$

Step 2.1.8.1.9

Rewrite $- 1 \cos^{2} (w)$ as $- \cos^{2} (w)$ .

$(\frac{\cos^{2} (w)}{\sin^{2} (w)} - \frac{\cos^{2} (w)}{\sin (w)} + \frac{\cos^{2} (w)}{\sin (w)} - \cos^{2} (w)) \sin^{2} (w) - \cos^{4} (w) = 0$

Step 2.1.8.2

Add $- \frac{\cos^{2} (w)}{\sin (w)}$ and $\frac{\cos^{2} (w)}{\sin (w)}$ .

$(\frac{\cos^{2} (w)}{\sin^{2} (w)} + 0 - \cos^{2} (w)) \sin^{2} (w) - \cos^{4} (w) = 0$

Step 2.1.8.3

Add $\frac{\cos^{2} (w)}{\sin^{2} (w)}$ and $0$ .

$(\frac{\cos^{2} (w)}{\sin^{2} (w)} - \cos^{2} (w)) \sin^{2} (w) - \cos^{4} (w) = 0$

Step 2.1.9

Apply the distributive property.

$\frac{\cos^{2} (w)}{\sin^{2} (w)} \sin^{2} (w) - \cos^{2} (w) \sin^{2} (w) - \cos^{4} (w) = 0$

Step 2.1.10

Cancel the common factor of $\sin^{2} (w)$ .

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Step 2.1.10.1

Cancel the common factor.

$\frac{\cos^{2} (w)}{\sin^{2} (w)} \sin^{2} (w) - \cos^{2} (w) \sin^{2} (w) - \cos^{4} (w) = 0$

Step 2.1.10.2

Rewrite the expression.

$\cos^{2} (w) - \cos^{2} (w) \sin^{2} (w) - \cos^{4} (w) = 0$

Step 2.1.11

Multiply by $1$ .

$\cos^{2} (w) \cdot 1 - \cos^{2} (w) \sin^{2} (w) - \cos^{4} (w) = 0$

Step 2.1.12

Factor $\cos^{2} (w)$ out of $- \cos^{2} (w) \sin^{2} (w)$ .

$\cos^{2} (w) \cdot 1 + \cos^{2} (w) (- 1 \sin^{2} (w)) - \cos^{4} (w) = 0$

Step 2.1.13

Factor $\cos^{2} (w)$ out of $\cos^{2} (w) \cdot 1 + \cos^{2} (w) (- 1 \sin^{2} (w))$ .

$\cos^{2} (w) \cdot (1 - 1 \sin^{2} (w)) - \cos^{4} (w) = 0$

Step 2.1.14

Rewrite $- 1 \sin^{2} (w)$ as $- \sin^{2} (w)$ .

$\cos^{2} (w) \cdot (1 - \sin^{2} (w)) - \cos^{4} (w) = 0$

Step 2.1.15

Apply pythagorean identity.

$\cos^{2} (w) \cdot \cos^{2} (w) - \cos^{4} (w) = 0$

Step 2.1.16

Multiply $\cos^{2} (w)$ by $\cos^{2} (w)$ by adding the exponents.

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Step 2.1.16.1

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

${\cos (w)}^{2 + 2} - \cos^{4} (w) = 0$

Step 2.1.16.2

Add $2$ and $2$ .

$\cos^{4} (w) - \cos^{4} (w) = 0$

Step 2.2

Subtract $\cos^{4} (w)$ from $\cos^{4} (w)$ .

$0 = 0$

Step 3

The domain of the expression is all real numbers except where the expression is undefined. In this case, there is no real number that makes the expression undefined.