Trigonometry Examples

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

Step 1

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

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

Step 2

Expand $(\csc^{2} (x) - 1) (\sec^{2} (x) - 1)$ using the FOIL Method.

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Step 2.1

Apply the distributive property.

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

Step 2.2

Apply the distributive property.

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

Step 2.3

Apply the distributive property.

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

Step 3

Simplify each term.

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Step 3.1

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

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

Step 3.2

Rewrite $- 1 \csc^{2} (x)$ as $- \csc^{2} (x)$ .

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

Step 3.3

Rewrite $- 1 \sec^{2} (x)$ as $- \sec^{2} (x)$ .

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

Step 3.4

Multiply $- 1$ by $- 1$ .

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

Step 4

Simplify the left side.

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Step 4.1

Simplify $\csc^{2} (x) \sec^{2} (x) - \csc^{2} (x) - \sec^{2} (x) + 1$ .

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Step 4.1.1

Simplify with factoring out.

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Step 4.1.1.1

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

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

Step 4.1.1.2

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

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

Step 4.1.1.3

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

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

Step 4.1.2

Apply pythagorean identity.

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

Step 4.1.3

Simplify with factoring out.

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Step 4.1.3.1

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

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

Step 4.1.3.2

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

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

Step 4.1.3.3

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

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

Step 4.1.4

Apply pythagorean identity.

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

Step 4.1.5

Rewrite $\csc^{2} (x) \tan^{2} (x)$ as ${(\csc (x) \tan (x))}^{2}$ .

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

Step 4.1.6

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

$(\csc (x) \tan (x) + \tan (x)) (\csc (x) \tan (x) - \tan (x)) = 1$

Step 4.1.7

Simplify terms.

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Step 4.1.7.1

Simplify each term.

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Step 4.1.7.1.1

Rewrite in terms of sines and cosines, then cancel the common factors.

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Step 4.1.7.1.1.1

Reorder $\csc (x)$ and $\tan (x)$ .

$(\tan (x) \csc (x) + \tan (x)) (\csc (x) \tan (x) - \tan (x)) = 1$

Step 4.1.7.1.1.2

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

$(\frac{\sin (x)}{\cos (x)} \cdot \frac{1}{\sin (x)} + \tan (x)) (\csc (x) \tan (x) - \tan (x)) = 1$

Step 4.1.7.1.1.3

Cancel the common factors.

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

Step 4.1.7.1.2

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

$(\sec (x) + \tan (x)) (\csc (x) \tan (x) - \tan (x)) = 1$

Step 4.1.7.2

Simplify each term.

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Step 4.1.7.2.1

Rewrite in terms of sines and cosines, then cancel the common factors.

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Step 4.1.7.2.1.1

Reorder $\csc (x)$ and $\tan (x)$ .

$(\sec (x) + \tan (x)) (\tan (x) \csc (x) - \tan (x)) = 1$

Step 4.1.7.2.1.2

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

$(\sec (x) + \tan (x)) (\frac{\sin (x)}{\cos (x)} \cdot \frac{1}{\sin (x)} - \tan (x)) = 1$

Step 4.1.7.2.1.3

Cancel the common factors.

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

Step 4.1.7.2.2

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

$(\sec (x) + \tan (x)) (\sec (x) - \tan (x)) = 1$

Step 4.1.8

Expand $(\sec (x) + \tan (x)) (\sec (x) - \tan (x))$ using the FOIL Method.

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Step 4.1.8.1

Apply the distributive property.

$\sec (x) (\sec (x) - \tan (x)) + \tan (x) (\sec (x) - \tan (x)) = 1$

Step 4.1.8.2

Apply the distributive property.

$\sec (x) \sec (x) + \sec (x) (- \tan (x)) + \tan (x) (\sec (x) - \tan (x)) = 1$

Step 4.1.8.3

Apply the distributive property.

$\sec (x) \sec (x) + \sec (x) (- \tan (x)) + \tan (x) \sec (x) + \tan (x) (- \tan (x)) = 1$

Step 4.1.9

Simplify terms.

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Step 4.1.9.1

Combine the opposite terms in $\sec (x) \sec (x) + \sec (x) (- \tan (x)) + \tan (x) \sec (x) + \tan (x) (- \tan (x))$ .

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Step 4.1.9.1.1

Reorder the factors in the terms $\sec (x) (- \tan (x))$ and $\tan (x) \sec (x)$ .

$\sec (x) \sec (x) - \sec (x) \tan (x) + \sec (x) \tan (x) + \tan (x) (- \tan (x)) = 1$

Step 4.1.9.1.2

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

$\sec (x) \sec (x) + 0 + \tan (x) (- \tan (x)) = 1$

Step 4.1.9.1.3

Add $\sec (x) \sec (x)$ and $0$ .

$\sec (x) \sec (x) + \tan (x) (- \tan (x)) = 1$

Step 4.1.9.2

Simplify each term.

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Step 4.1.9.2.1

Multiply $\sec (x) \sec (x)$ .

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Step 4.1.9.2.1.1

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

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

Step 4.1.9.2.1.2

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

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

Step 4.1.9.2.1.3

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

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

Step 4.1.9.2.1.4

Add $1$ and $1$ .

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

Step 4.1.9.2.2

Rewrite using the commutative property of multiplication.

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

Step 4.1.9.2.3

Multiply $- \tan (x) \tan (x)$ .

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Step 4.1.9.2.3.1

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

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

Step 4.1.9.2.3.2

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

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

Step 4.1.9.2.3.3

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

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

Step 4.1.9.2.3.4

Add $1$ and $1$ .

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

Step 4.1.10

Apply pythagorean identity.

$1 = 1$

Step 5

Since $1 = 1$ , the equation will always be true for any value of $x$ .

All real numbers

Step 6

The result can be shown in multiple forms.

All real numbers

Interval Notation:

$(- \infty, \infty)$