Trigonometry Examples

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

Step 1

Start on the left side.

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

Step 2

Convert to sines and cosines.

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Step 2.1

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

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

Step 2.2

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

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

Step 3

Simplify.

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Step 3.1

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

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Step 3.1.1

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

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

Step 3.1.2

Combine.

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

Step 3.2

Apply the distributive property.

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

Step 3.3

Simplify by cancelling.

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Step 3.3.1

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

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Step 3.3.1.1

Cancel the common factor.

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

Step 3.3.1.2

Rewrite the expression.

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

Step 3.3.2

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

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Step 3.3.2.1

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

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

Step 3.3.2.2

Cancel the common factor.

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

Step 3.3.2.3

Rewrite the expression.

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

Step 3.4

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

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

Step 3.5

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

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

Step 4

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

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

Step 5

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

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