Trigonometry Examples

tan (4 x)

$\tan (4 x)$

Step 1

Factor

2

$2$ out of

4 x

$4 x$ .

tan (2 (2 x))

$\tan (2 (2 x))$

Step 2

Simplify the numerator.

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Step 2.1

Apply the tangent double-angle identity.

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

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

Step 2.2

Simplify the denominator.

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Step 2.2.1

Rewrite

1

$1$ as

1^{2}

$1^{2}$ .

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

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

Step 2.2.2

Since both terms are perfect squares, factor using the difference of squares formula,

a^{2} - b^{2} = (a + b) (a - b)

$a^{2} - b^{2} = (a + b) (a - b)$ where

a = 1

$a = 1$ and

b = tan (x)

$b = \tan (x)$ .

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

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

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

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

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

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

Step 3

Simplify the denominator.

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Step 3.1

Rewrite

1

$1$ as

1^{2}

$1^{2}$ .

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

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

Step 3.2

Since both terms are perfect squares, factor using the difference of squares formula,

a^{2} - b^{2} = (a + b) (a - b)

$a^{2} - b^{2} = (a + b) (a - b)$ where

a = 1

$a = 1$ and

b = tan (2 x)

$b = \tan (2 x)$ .

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

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

Step 3.3

Simplify.

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Step 3.3.1

Apply the tangent double-angle identity.

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

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

Step 3.3.2

Simplify the denominator.

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Step 3.3.2.1

Rewrite

1

$1$ as

1^{2}

$1^{2}$ .

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

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

Step 3.3.2.2

Since both terms are perfect squares, factor using the difference of squares formula,

a^{2} - b^{2} = (a + b) (a - b)

$a^{2} - b^{2} = (a + b) (a - b)$ where

a = 1

$a = 1$ and

b = tan (x)

$b = \tan (x)$ .

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

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

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

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

Step 3.3.3

Apply the tangent double-angle identity.

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

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

Step 3.3.4

Simplify the denominator.

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Step 3.3.4.1

Rewrite

1

$1$ as

1^{2}

$1^{2}$ .

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

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

Step 3.3.4.2

Since both terms are perfect squares, factor using the difference of squares formula,

a^{2} - b^{2} = (a + b) (a - b)

$a^{2} - b^{2} = (a + b) (a - b)$ where

a = 1

$a = 1$ and

b = tan (x)

$b = \tan (x)$ .

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

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

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

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

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

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

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

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

Step 4

Combine

2

$2$ and

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

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

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

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

Step 5

Multiply

2

$2$ by

2

$2$ .

\frac{\frac{4 tan (x)}{(1 + tan (x)) (1 - tan (x))}}{(1 + \frac{2 tan (x)}{(1 + tan (x)) (1 - tan (x))}) (1 - \frac{2 tan (x)}{(1 + tan (x)) (1 - tan (x))})}

$\frac{\frac{4 \tan (x)}{(1 + \tan (x)) (1 - \tan (x))}}{(1 + \frac{2 \tan (x)}{(1 + \tan (x)) (1 - \tan (x))}) (1 - \frac{2 \tan (x)}{(1 + \tan (x)) (1 - \tan (x))})}$

Step 6

Multiply the numerator by the reciprocal of the denominator.

\frac{4 tan (x)}{(1 + tan (x)) (1 - tan (x))} \cdot \frac{1}{(1 + \frac{2 tan (x)}{(1 + tan (x)) (1 - tan (x))}) (1 - \frac{2 tan (x)}{(1 + tan (x)) (1 - tan (x))})}

$\frac{4 \tan (x)}{(1 + \tan (x)) (1 - \tan (x))} \cdot \frac{1}{(1 + \frac{2 \tan (x)}{(1 + \tan (x)) (1 - \tan (x))}) (1 - \frac{2 \tan (x)}{(1 + \tan (x)) (1 - \tan (x))})}$

Step 7

Multiply

\frac{4 tan (x)}{(1 + tan (x)) (1 - tan (x))}

$\frac{4 \tan (x)}{(1 + \tan (x)) (1 - \tan (x))}$ by

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

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

\frac{4 tan (x)}{(1 + tan (x)) (1 - tan (x)) (1 + \frac{2 tan (x)}{(1 + tan (x)) (1 - tan (x))}) (1 - \frac{2 tan (x)}{(1 + tan (x)) (1 - tan (x))})}

$\frac{4 \tan (x)}{(1 + \tan (x)) (1 - \tan (x)) (1 + \frac{2 \tan (x)}{(1 + \tan (x)) (1 - \tan (x))}) (1 - \frac{2 \tan (x)}{(1 + \tan (x)) (1 - \tan (x))})}$