Trigonometry Examples

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

Step 1

Start on the left side.

$\sec^{2} (x) + \cot^{2} (x)$

Step 2

Apply Pythagorean identity in reverse.

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

Step 3

Convert to sines and cosines.

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Step 3.1

Write $\tan (x)$ in sines and cosines using the quotient identity.

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

Step 3.2

Write $\cot (x)$ in sines and cosines using the quotient identity.

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

Step 3.3

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

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

Step 3.4

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

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

Step 4

Write $\frac{\sin^{2} (x)}{\cos^{2} (x)} + 1$ as a fraction with denominator $1$ .

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

Step 5

Add fractions.

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Step 5.1

To write $\frac{\frac{\sin^{2} (x)}{\cos^{2} (x)} + 1}{1}$ as a fraction with a common denominator, multiply by $\frac{\sin^{2} (x)}{\sin^{2} (x)}$ .

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

Step 5.2

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

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

Step 5.3

Combine the numerators over the common denominator.

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

Step 6

Simplify each term.

$\frac{\frac{\sin^{4} (x)}{\cos^{2} (x)} + \sin^{2} (x) + \cos^{2} (x)}{\sin^{2} (x)}$

Step 7

Write $\sin^{2} (x)$ as a fraction with denominator $1$ .

$\frac{\frac{\sin^{4} (x)}{\cos^{2} (x)} + \frac{\sin^{2} (x)}{1} + \cos^{2} (x)}{\sin^{2} (x)}$

Step 8

Add fractions.

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Step 8.1

To write $\frac{\sin^{2} (x)}{1}$ as a fraction with a common denominator, multiply by $\frac{\cos^{2} (x)}{\cos^{2} (x)}$ .

$\frac{\frac{\sin^{4} (x)}{\cos^{2} (x)} + \frac{\sin^{2} (x)}{1} \cdot \frac{\cos^{2} (x)}{\cos^{2} (x)} + \cos^{2} (x)}{\sin^{2} (x)}$

Step 8.2

Multiply $\frac{\sin^{2} (x)}{1}$ by $\frac{\cos^{2} (x)}{\cos^{2} (x)}$ .

$\frac{\frac{\sin^{4} (x)}{\cos^{2} (x)} + \frac{\sin^{2} (x) \cos^{2} (x)}{\cos^{2} (x)} + \cos^{2} (x)}{\sin^{2} (x)}$

Step 8.3

Combine the numerators over the common denominator.

$\frac{\frac{\sin^{4} (x) + \sin^{2} (x) \cos^{2} (x)}{\cos^{2} (x)} + \cos^{2} (x)}{\sin^{2} (x)}$

Step 9

Write $\cos^{2} (x)$ as a fraction with denominator $1$ .

$\frac{\frac{\sin^{4} (x) + \sin^{2} (x) \cos^{2} (x)}{\cos^{2} (x)} + \frac{\cos^{2} (x)}{1}}{\sin^{2} (x)}$

Step 10

Add fractions.

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Step 10.1

To write $\frac{\cos^{2} (x)}{1}$ as a fraction with a common denominator, multiply by $\frac{\cos^{2} (x)}{\cos^{2} (x)}$ .

$\frac{\frac{\sin^{4} (x) + \sin^{2} (x) \cos^{2} (x)}{\cos^{2} (x)} + \frac{\cos^{2} (x)}{1} \cdot \frac{\cos^{2} (x)}{\cos^{2} (x)}}{\sin^{2} (x)}$

Step 10.2

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

$\frac{\frac{\sin^{4} (x) + \sin^{2} (x) \cos^{2} (x)}{\cos^{2} (x)} + \frac{\cos^{2} (x) \cos^{2} (x)}{\cos^{2} (x)}}{\sin^{2} (x)}$

Step 10.3

Combine the numerators over the common denominator.

$\frac{\frac{\sin^{4} (x) + \sin^{2} (x) \cos^{2} (x) + \cos^{2} (x) \cos^{2} (x)}{\cos^{2} (x)}}{\sin^{2} (x)}$

Step 11

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

$\frac{\frac{\sin^{4} (x) + \sin^{2} (x) \cos^{2} (x) + \cos^{4} (x)}{\cos^{2} (x)}}{\sin^{2} (x)}$

Step 12

Apply Pythagorean identity in reverse.

$\frac{\frac{\sin^{4} (x) + (1 - \cos^{2} (x)) \cos^{2} (x) + \cos^{4} (x)}{\cos^{2} (x)}}{\sin^{2} (x)}$

Step 13

Simplify.

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Step 13.1

Multiply the numerator by the reciprocal of the denominator.

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

Step 13.2

Simplify the numerator.

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Step 13.2.1

Apply the distributive property.

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

Step 13.2.2

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

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

Step 13.2.3

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

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Step 13.2.3.1

Move ${\cos (x)}^{2}$ .

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

Step 13.2.3.2

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

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

Step 13.2.3.3

Add $2$ and $2$ .

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

Step 13.2.4

Add $- {\cos (x)}^{4}$ and ${\cos (x)}^{4}$ .

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

Step 13.2.5

Add ${\sin (x)}^{4} + {\cos (x)}^{2}$ and $0$ .

$\frac{{\sin (x)}^{4} + {\cos (x)}^{2}}{{\cos (x)}^{2}} \cdot \frac{1}{{\sin (x)}^{2}}$

Step 13.3

Multiply $\frac{{\sin (x)}^{4} + {\cos (x)}^{2}}{{\cos (x)}^{2}}$ by $\frac{1}{{\sin (x)}^{2}}$ .

$\frac{\sin^{4} (x) + \cos^{2} (x)}{\cos^{2} (x) \sin^{2} (x)}$

Step 14

Now consider the right side of the equation.

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

Step 15

Convert to sines and cosines.

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Step 15.1

Write $\tan (x)$ in sines and cosines using the quotient identity.

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

Step 15.2

Apply the reciprocal identity to $\csc (x)$ .

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

Step 15.3

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

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

Step 15.4

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

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

Step 16

One to any power is one.

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

Step 17

Add fractions.

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Step 17.1

To write $\frac{\sin^{2} (x)}{\cos^{2} (x)}$ as a fraction with a common denominator, multiply by $\frac{\sin^{2} (x)}{\sin^{2} (x)}$ .

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

Step 17.2

To write $\frac{1}{\sin^{2} (x)}$ as a fraction with a common denominator, multiply by $\frac{\cos^{2} (x)}{\cos^{2} (x)}$ .

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

Step 17.3

Write each expression with a common denominator of $\cos^{2} (x) \sin^{2} (x)$ , by multiplying each by an appropriate factor of $1$ .

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Step 17.3.1

Multiply $\frac{\sin^{2} (x)}{\cos^{2} (x)}$ by $\frac{\sin^{2} (x)}{\sin^{2} (x)}$ .

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

Step 17.3.2

Multiply $\frac{1}{\sin^{2} (x)}$ by $\frac{\cos^{2} (x)}{\cos^{2} (x)}$ .

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

Step 17.3.3

Reorder the factors of $\sin^{2} (x) \cos^{2} (x)$ .

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

Step 17.4

Combine the numerators over the common denominator.

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

Step 18

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

$\frac{\sin^{4} (x) + \cos^{2} (x)}{\cos^{2} (x) \sin^{2} (x)}$

Step 19

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

$\sec^{2} (x) + \cot^{2} (x) = \tan^{2} (x) + \csc^{2} (x)$ is an identity