Trigonometry Examples

$\cot (45)$

Step 1

Replace $\cot (45)$ with an equivalent expression $\frac{1}{\tan (45)}$ using the fundamental identities.

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

Step 2

Use a sum or difference formula on the denominator.

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Step 2.1

The angle $45$ is an angle where the values of the six trigonometric functions are known. Because this is the case, add $0$ to keep the value the same.

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

Step 2.2

Use the sum formula for tangent to simplify the expression. The formula states that $\tan (A + B) = \frac{\tan (A) + \tan (B)}{1 - \tan (A) \tan (B)}$ .

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

Step 2.3

Simplify the numerator.

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Step 2.3.1

The exact value of $\tan (45)$ is $1$ .

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

Step 2.3.2

The exact value of $\tan (0)$ is $0$ .

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

Step 2.3.3

Add $1$ and $0$ .

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

Step 2.4

Simplify the denominator.

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Step 2.4.1

The exact value of $\tan (45)$ is $1$ .

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

Step 2.4.2

Multiply $- 1$ by $1$ .

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

Step 2.4.3

The exact value of $\tan (0)$ is $0$ .

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

Step 2.4.4

Multiply $- 1$ by $0$ .

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

Step 2.4.5

Add $1$ and $0$ .

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

Step 2.5

Cancel the common factor of $1$ .

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Step 2.5.1

Cancel the common factor.

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

Step 2.5.2

Rewrite the expression.

$\frac{1}{1}$

Step 3

Divide $1$ by $1$ .

$1$