Trigonometry Examples

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

Step 1

Start on the right side.

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

Step 2

Simplify the expression.

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Step 2.1

Rearrange terms.

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

Step 2.2

Apply pythagorean identity.

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

Step 2.3

Simplify the numerator.

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Step 2.3.1

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

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

Step 2.3.2

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 $b = \tan (x)$ .

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

Step 2.3.3

Simplify.

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Step 2.3.3.1

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

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

Step 2.3.3.2

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

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

Step 2.4

Simplify the denominator.

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Step 2.4.1

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

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

Step 2.4.2

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

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

Step 2.4.3

One to any power is one.

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

Step 2.5

Multiply the numerator by the reciprocal of the denominator.

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

Step 2.6

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

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Step 2.6.1

Apply the distributive property.

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

Step 2.6.2

Apply the distributive property.

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

Step 2.6.3

Apply the distributive property.

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

Step 2.7

Simplify and combine like terms.

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Step 2.7.1

Simplify each term.

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Step 2.7.1.1

Multiply $1$ by $1$ .

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

Step 2.7.1.2

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

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

Step 2.7.1.3

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

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

Step 2.7.1.4

Rewrite using the commutative property of multiplication.

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

Step 2.7.1.5

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

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Step 2.7.1.5.1

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

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

Step 2.7.1.5.2

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

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

Step 2.7.1.5.3

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

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

Step 2.7.1.5.4

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

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

Step 2.7.1.5.5

Add $1$ and $1$ .

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

Step 2.7.1.5.6

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

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

Step 2.7.1.5.7

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

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

Step 2.7.1.5.8

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

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

Step 2.7.1.5.9

Add $1$ and $1$ .

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

Step 2.7.2

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

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

Step 2.7.3

Add $1$ and $0$ .

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

Step 2.8

Apply the distributive property.

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

Step 2.9

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

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

Step 2.10

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

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Step 2.10.1

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

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

Step 2.10.2

Cancel the common factor.

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

Step 2.10.3

Rewrite the expression.

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

Step 2.11

Apply the cosine double-angle identity.

$\cos (2 x)$

Step 3

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

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