Trigonometry Examples

\cot (45)

$\cot (45)$

Step 1

Replace

\cot (45)

$\cot (45)$ with an equivalent expression

\frac{1}{\tan (45)}

$\frac{1}{\tan (45)}$ using the fundamental identities.

\frac{1}{\tan (45)}

$\frac{1}{\tan (45)}$

Step 2

Use a sum or difference formula on the denominator.

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Step 2.1

The angle

45

$45$ is an angle where the values of the six trigonometric functions are known. Because this is the case, add

0

$0$ to keep the value the same.

\frac{1}{\tan (45 + 0)}

$\frac{1}{\tan (45 + 0)}$

Step 2.2

Use the sum formula for tangent to simplify the expression. The formula states that

\tan (A + B) = \frac{\tan (A) + \tan (B)}{1 - \tan (A) \tan (B)}

$\tan (A + B) = \frac{\tan (A) + \tan (B)}{1 - \tan (A) \tan (B)}$ .

\frac{1}{\frac{\tan (45) + \tan (0)}{1 - \tan (45) \tan (0)}}

$\frac{1}{\frac{\tan (45) + \tan (0)}{1 - \tan (45) \tan (0)}}$

Step 2.3

Simplify the numerator.

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Step 2.3.1

The exact value of

\tan (45)

$\tan (45)$ is

1

$1$ .

\frac{1}{\frac{1 + \tan (0)}{1 - \tan (45) \tan (0)}}

$\frac{1}{\frac{1 + \tan (0)}{1 - \tan (45) \tan (0)}}$

Step 2.3.2

The exact value of

\tan (0)

$\tan (0)$ is

0

$0$ .

\frac{1}{\frac{1 + 0}{1 - \tan (45) \tan (0)}}

$\frac{1}{\frac{1 + 0}{1 - \tan (45) \tan (0)}}$

Step 2.3.3

Add

1

$1$ and

0

$0$ .

\frac{1}{\frac{1}{1 - \tan (45) \tan (0)}}

$\frac{1}{\frac{1}{1 - \tan (45) \tan (0)}}$

\frac{1}{\frac{1}{1 - \tan (45) \tan (0)}}

$\frac{1}{\frac{1}{1 - \tan (45) \tan (0)}}$

Step 2.4

Simplify the denominator.

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Step 2.4.1

The exact value of

\tan (45)

$\tan (45)$ is

1

$1$ .

\frac{1}{\frac{1}{1 - 1 \cdot (1 \tan (0))}}

$\frac{1}{\frac{1}{1 - 1 \cdot (1 \tan (0))}}$

Step 2.4.2

Multiply

- 1

$- 1$ by

1

$1$ .

\frac{1}{\frac{1}{1 - 1 \tan (0)}}

$\frac{1}{\frac{1}{1 - 1 \tan (0)}}$

Step 2.4.3

The exact value of

\tan (0)

$\tan (0)$ is

0

$0$ .

\frac{1}{\frac{1}{1 - 1 \cdot 0}}

$\frac{1}{\frac{1}{1 - 1 \cdot 0}}$

Step 2.4.4

Multiply

- 1

$- 1$ by

0

$0$ .

\frac{1}{\frac{1}{1 + 0}}

$\frac{1}{\frac{1}{1 + 0}}$

Step 2.4.5

Add

1

$1$ and

0

$0$ .

\frac{1}{\frac{1}{1}}

$\frac{1}{\frac{1}{1}}$

\frac{1}{\frac{1}{1}}

$\frac{1}{\frac{1}{1}}$

Step 2.5

Cancel the common factor of

1

$1$ .

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Step 2.5.1

Cancel the common factor.

\frac{1}{\frac{1}{1}}

$\frac{1}{\frac{1}{1}}$

Step 2.5.2

Rewrite the expression.

\frac{1}{1}

$\frac{1}{1}$

\frac{1}{1}

$\frac{1}{1}$

\frac{1}{1}

$\frac{1}{1}$

Step 3

Divide

1

$1$ by

1

$1$ .

1

$1$