Trigonometry Examples

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

Step 1

Start on the left side.

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

Step 2

Convert to sines and cosines.

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Step 2.1

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

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

Step 2.2

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

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

Step 3

Simplify.

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Step 3.1

Multiply the numerator and denominator of the fraction by ${\cos (x)}^{2}$ .

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Step 3.1.1

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

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

Step 3.1.2

Combine.

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

Step 3.2

Apply the distributive property.

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

Step 3.3

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

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Step 3.3.1

Move the leading negative in $- \frac{{\sin (x)}^{2}}{{\cos (x)}^{2}}$ into the numerator.

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

Step 3.3.2

Cancel the common factor.

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

Step 3.3.3

Rewrite the expression.

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

Step 3.4

Simplify the numerator.

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Step 3.4.1

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

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

Step 3.4.2

Rewrite ${\cos (x)}^{2} ({\sin (x)}^{2})$ as ${(\cos (x) \sin (x))}^{2}$ .

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

Step 3.4.3

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

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

Step 3.4.4

Simplify.

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Step 3.4.4.1

Multiply $\cos (x) \cos (x)$ .

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Step 3.4.4.1.1

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

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

Step 3.4.4.1.2

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

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

Step 3.4.4.1.3

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

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

Step 3.4.4.1.4

Add $1$ and $1$ .

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

Step 3.4.4.2

Factor $\cos (x)$ out of ${\cos (x)}^{2} + \cos (x) \sin (x)$ .

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Step 3.4.4.2.1

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

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

Step 3.4.4.2.2

Factor $\cos (x)$ out of $\cos (x) \sin (x)$ .

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

Step 3.4.4.2.3

Factor $\cos (x)$ out of $\cos (x) \cos (x) + \cos (x) (\sin (x))$ .

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

Step 3.4.4.3

Combine exponents.

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Step 3.4.4.3.1

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

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

Step 3.4.4.3.2

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

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

Step 3.4.4.3.3

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

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

Step 3.4.4.3.4

Add $1$ and $1$ .

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

Step 3.4.5

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

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Step 3.4.5.1

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

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

Step 3.4.5.2

Factor $\cos (x)$ out of $- \cos (x) \sin (x)$ .

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

Step 3.4.5.3

Factor $\cos (x)$ out of $\cos (x) \cos (x) + \cos (x) (- \sin (x))$ .

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

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

Step 3.4.6

Combine exponents.

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Step 3.4.6.1

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

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

Step 3.4.6.2

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

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

Step 3.4.6.3

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

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

Step 3.4.6.4

Add $1$ and $1$ .

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

Step 3.5

Simplify the denominator.

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Step 3.5.1

Rewrite ${\cos (x)}^{2} \cdot 1$ as ${(\cos (x) \cdot 1)}^{2}$ .

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

Step 3.5.2

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

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

Step 3.5.3

Simplify.

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Step 3.5.3.1

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

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

Step 3.5.3.2

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

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

Step 3.6

Cancel the common factor of $\cos (x) + \sin (x)$ .

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Step 3.6.1

Cancel the common factor.

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

Step 3.6.2

Rewrite the expression.

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

Step 3.7

Cancel the common factor of $\cos (x) - \sin (x)$ .

$\cos^{2} (x)$

Step 4

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

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