Trigonometry Examples

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

Step 1

Start on the left side.

$(\csc (x) + \cot (x)) (\csc (x) - \cot (x)) + \tan^{2} (x)$

Step 2

Simplify each term.

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Step 2.1

Simplify each term.

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Step 2.1.1

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

$(\frac{1}{\sin (x)} + \cot (x)) (\csc (x) - \cot (x)) + \tan^{2} (x)$

Step 2.1.2

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

$(\frac{1}{\sin (x)} + \frac{\cos (x)}{\sin (x)}) (\csc (x) - \cot (x)) + \tan^{2} (x)$

Step 2.2

Simplify each term.

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Step 2.2.1

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

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

Step 2.2.2

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

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

Step 2.3

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

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Step 2.3.1

Apply the distributive property.

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

Step 2.3.2

Apply the distributive property.

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

Step 2.3.3

Apply the distributive property.

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

Step 2.4

Combine the opposite terms in $\frac{1}{\sin (x)} \cdot \frac{1}{\sin (x)} + \frac{1}{\sin (x)} (- \frac{\cos (x)}{\sin (x)}) + \frac{\cos (x)}{\sin (x)} \cdot \frac{1}{\sin (x)} + \frac{\cos (x)}{\sin (x)} (- \frac{\cos (x)}{\sin (x)})$ .

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Step 2.4.1

Reorder the factors in the terms $\frac{1}{\sin (x)} (- \frac{\cos (x)}{\sin (x)})$ and $\frac{\cos (x)}{\sin (x)} \cdot \frac{1}{\sin (x)}$ .

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

Step 2.4.2

Add $- \frac{1}{\sin (x)} \cdot \frac{\cos (x)}{\sin (x)}$ and $\frac{1}{\sin (x)} \cdot \frac{\cos (x)}{\sin (x)}$ .

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

Step 2.4.3

Add $\frac{1}{\sin (x)} \cdot \frac{1}{\sin (x)}$ and $0$ .

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

Step 2.5

Simplify each term.

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Step 2.5.1

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

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Step 2.5.1.1

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

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

Step 2.5.1.2

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

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

Step 2.5.1.3

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

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

Step 2.5.1.4

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

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

Step 2.5.1.5

Add $1$ and $1$ .

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

Step 2.5.2

Rewrite using the commutative property of multiplication.

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

Step 2.5.3

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

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Step 2.5.3.1

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

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

Step 2.5.3.2

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

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

Step 2.5.3.3

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

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

Step 2.5.3.4

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

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

Step 2.5.3.5

Add $1$ and $1$ .

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

Step 2.5.3.6

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

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

Step 2.5.3.7

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

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

Step 2.5.3.8

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

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

Step 2.5.3.9

Add $1$ and $1$ .

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

Step 2.6

Combine the numerators over the common denominator.

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

Step 2.7

Apply pythagorean identity.

$\frac{\sin^{2} (x)}{\sin^{2} (x)} + \tan^{2} (x)$

Step 2.8

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

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Step 2.8.1

Cancel the common factor.

$\frac{\sin^{2} (x)}{\sin^{2} (x)} + \tan^{2} (x)$

Step 2.8.2

Rewrite the expression.

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

Step 2.9

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

$1 + {(\frac{\sin (x)}{\cos (x)})}^{2}$

Step 2.10

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

$1 + \frac{\sin^{2} (x)}{\cos^{2} (x)}$

Step 3

Apply Pythagorean identity in reverse.

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

Step 4

Simplify.

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Step 4.1

Simplify the numerator.

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Step 4.1.1

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

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

Step 4.1.2

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

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

Step 4.2

Write $1$ as a fraction with a common denominator.

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

Step 4.3

Combine the numerators over the common denominator.

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

Step 4.4

Simplify the numerator.

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

Step 5

Because the two sides have been shown to be equivalent, the equation is an identity.

$(\csc (x) + \cot (x)) (\csc (x) - \cot (x)) + \tan^{2} (x) = \frac{1}{\cos^{2} (x)}$ is an identity