Trigonometry Examples

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

Step 1

Start on the right side.

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

Step 2

Apply Pythagorean identity in reverse.

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

Step 3

Factor.

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Step 3.1

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

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

Step 3.2

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 $b = \sin (x)$ .

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

Step 4

Apply Pythagorean identity in reverse.

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

Step 5

Convert to sines and cosines.

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Step 5.1

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

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

Step 5.2

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

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

Step 6

Simplify.

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Step 6.1

Simplify each term.

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Step 6.1.1

Expand $(1 + \sin (x)) (1 - \sin (x))$ using the FOIL Method.

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Step 6.1.1.1

Apply the distributive property.

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

Step 6.1.1.2

Apply the distributive property.

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

Step 6.1.1.3

Apply the distributive property.

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

Step 6.1.2

Simplify and combine like terms.

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Step 6.1.2.1

Simplify each term.

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Step 6.1.2.1.1

Multiply $1$ by $1$ .

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

Step 6.1.2.1.2

Multiply $- \sin (x)$ by $1$ .

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

Step 6.1.2.1.3

Multiply $\sin (x)$ by $1$ .

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

Step 6.1.2.1.4

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

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Step 6.1.2.1.4.1

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

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

Step 6.1.2.1.4.2

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

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

Step 6.1.2.1.4.3

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

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

Step 6.1.2.1.4.4

Add $1$ and $1$ .

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

Step 6.1.2.2

Add $- \sin (x)$ and $\sin (x)$ .

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

Step 6.1.2.3

Add $1$ and $0$ .

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

Step 6.1.3

One to any power is one.

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

Step 6.2

Subtract $1$ from $1$ .

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

Step 6.3

Add $- {\sin (x)}^{2}$ and $0$ .

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

Step 7

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

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

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

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

Step 8.2

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

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

Step 8.3

Combine the numerators over the common denominator.

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

Step 9

Simplify the numerator.

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

Step 10

Now consider the left side of the equation.

$\sec^{2} (x) - \sin^{2} (x)$

Step 11

Convert to sines and cosines.

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Step 11.1

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

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

Step 11.2

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

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

Step 12

One to any power is one.

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

Step 13

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

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

Step 14

Add fractions.

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Step 14.1

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

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

Step 14.2

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

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

Step 14.3

Combine the numerators over the common denominator.

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

Step 15

Simplify the numerator.

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

Step 16

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

$\sec^{2} (x) - \sin^{2} (x) = \cos^{2} (x) + \tan^{2} (x)$ is an identity