Trigonometry Examples

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

Step 1

Start on the left side.

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

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.

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

Step 2.2

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

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

Step 3

Simplify.

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Step 3.1

Multiply the numerator and denominator of the fraction by $\cos (x)$ .

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Step 3.1.1

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

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

Step 3.1.2

Combine.

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

Step 3.2

Apply the distributive property.

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

Step 3.3

Simplify by cancelling.

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Step 3.3.1

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

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Step 3.3.1.1

Cancel the common factor.

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

Step 3.3.1.2

Rewrite the expression.

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

Step 3.3.2

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

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Step 3.3.2.1

Cancel the common factor.

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

Step 3.3.2.2

Rewrite the expression.

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

Step 3.4

Simplify the numerator.

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Step 3.4.1

Move $- 1$ to the left of $\cos (x)$ .

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

Step 3.4.2

Rewrite $- 1 \cos (x)$ as $- \cos (x)$ .

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

Step 3.5

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

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

Step 4

Reorder terms.

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

Step 5

Rewrite $\frac{\sin (x) - \cos (x)}{\cos (x) + \sin (x)}$ as $\frac{1 - \cot (x)}{1 + \cot (x)}$ .

$\frac{1 - \cot (x)}{1 + \cot (x)}$

Step 6

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

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