Trigonometry Examples

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

Step 1

Multiply both sides by $\tan (x) + 1$ .

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

Step 2

Simplify.

Tap for more steps...

Step 2.1

Simplify the left side.

Tap for more steps...

Step 2.1.1

Cancel the common factor of $\tan (x) + 1$ .

Tap for more steps...

Step 2.1.1.1

Cancel the common factor.

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

Step 2.1.1.2

Rewrite the expression.

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

Step 2.2

Simplify the right side.

Tap for more steps...

Step 2.2.1

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

Tap for more steps...

Step 2.2.1.1

Simplify terms.

Tap for more steps...

Step 2.2.1.1.1

Apply the distributive property.

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

Step 2.2.1.1.2

Combine $\frac{1 - \cot (x)}{1 + \cot (x)}$ and $\tan (x)$ .

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

Step 2.2.1.1.3

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

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

Step 2.2.1.1.4

Combine the numerators over the common denominator.

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

Step 2.2.1.2

Simplify each term.

Tap for more steps...

Step 2.2.1.2.1

Apply the distributive property.

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

Step 2.2.1.2.2

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

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

Step 2.2.1.2.3

Rewrite in terms of sines and cosines, then cancel the common factors.

Tap for more steps...

Step 2.2.1.2.3.1

Add parentheses.

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

Step 2.2.1.2.3.2

Rewrite $- \cot (x) \tan (x)$ in terms of sines and cosines.

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

Step 2.2.1.2.3.3

Cancel the common factors.

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

Step 2.2.1.2.4

Multiply $- 1$ by $1$ .

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

Step 2.2.1.3

Combine the opposite terms in $\tan (x) - 1 + 1 - \cot (x)$ .

Tap for more steps...

Step 2.2.1.3.1

Add $- 1$ and $1$ .

$\tan (x) - 1 = \frac{\tan (x) + 0 - \cot (x)}{1 + \cot (x)}$

Step 2.2.1.3.2

Add $\tan (x)$ and $0$ .

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

Step 3

Solve for $x$ .

Tap for more steps...

Step 3.1

Simplify the right side.

Tap for more steps...

Step 3.1.1

Combine the numerators over the common denominator.

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

Step 3.2

Simplify the left side.

Tap for more steps...

Step 3.2.1

Rewrite $\tan (x)$ in terms of sines and cosines.

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

Step 3.3

Simplify the right side.

Tap for more steps...

Step 3.3.1

Simplify each term.

Tap for more steps...

Step 3.3.1.1

Simplify the numerator.

Tap for more steps...

Step 3.3.1.1.1

Rewrite $\tan (x)$ in terms of sines and cosines.

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

Step 3.3.1.1.2

Rewrite $\cot (x)$ in terms of sines and cosines.

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

Step 3.3.1.2

Rewrite $\cot (x)$ in terms of sines and cosines.

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

Step 3.3.1.3

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

Tap for more steps...

Step 3.3.1.3.1

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

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

Step 3.3.1.3.2

Combine.

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

Step 3.3.1.4

Apply the distributive property.

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

Step 3.3.1.5

Simplify by cancelling.

Tap for more steps...

Step 3.3.1.5.1

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

Tap for more steps...

Step 3.3.1.5.1.1

Factor $\cos (x)$ out of $\cos (x) \sin (x)$ .

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

Step 3.3.1.5.1.2

Cancel the common factor.

Step 3.3.1.5.1.3

Rewrite the expression.

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

Step 3.3.1.5.2

Raise $\sin (x)$ to the power of $1$ .

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

Step 3.3.1.5.3

Raise $\sin (x)$ to the power of $1$ .

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

Step 3.3.1.5.4

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Step 3.3.1.5.5

Add $1$ and $1$ .

$\frac{\sin (x)}{\cos (x)} = \frac{\sin^{2} (x) + \cos (x) \sin (x) (- \frac{\cos (x)}{\sin (x)})}{\cos (x) \sin (x) \cdot 1 + \cos (x) \sin (x) \frac{\cos (x)}{\sin (x)}} + 1$

Step 3.3.1.5.6

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

Tap for more steps...

Step 3.3.1.5.6.1

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

$\frac{\sin (x)}{\cos (x)} = \frac{\sin^{2} (x) + \cos (x) \sin (x) \frac{- \cos (x)}{\sin (x)}}{\cos (x) \sin (x) \cdot 1 + \cos (x) \sin (x) \frac{\cos (x)}{\sin (x)}} + 1$

Step 3.3.1.5.6.2

Factor $\sin (x)$ out of $\cos (x) \sin (x)$ .

$\frac{\sin (x)}{\cos (x)} = \frac{\sin^{2} (x) + \sin (x) \cos (x) \frac{- \cos (x)}{\sin (x)}}{\cos (x) \sin (x) \cdot 1 + \cos (x) \sin (x) \frac{\cos (x)}{\sin (x)}} + 1$

Step 3.3.1.5.6.3

Cancel the common factor.

$\frac{\sin (x)}{\cos (x)} = \frac{\sin^{2} (x) + \sin (x) \cos (x) \frac{- \cos (x)}{\sin (x)}}{\cos (x) \sin (x) \cdot 1 + \cos (x) \sin (x) \frac{\cos (x)}{\sin (x)}} + 1$

Step 3.3.1.5.6.4

Rewrite the expression.

$\frac{\sin (x)}{\cos (x)} = \frac{\sin^{2} (x) + \cos (x) (- \cos (x))}{\cos (x) \sin (x) \cdot 1 + \cos (x) \sin (x) \frac{\cos (x)}{\sin (x)}} + 1$

Step 3.3.1.5.7

Raise $\cos (x)$ to the power of $1$ .

$\frac{\sin (x)}{\cos (x)} = \frac{\sin^{2} (x) - (\cos^{1} (x) \cos (x))}{\cos (x) \sin (x) \cdot 1 + \cos (x) \sin (x) \frac{\cos (x)}{\sin (x)}} + 1$

Step 3.3.1.5.8

Raise $\cos (x)$ to the power of $1$ .

$\frac{\sin (x)}{\cos (x)} = \frac{\sin^{2} (x) - (\cos^{1} (x) \cos^{1} (x))}{\cos (x) \sin (x) \cdot 1 + \cos (x) \sin (x) \frac{\cos (x)}{\sin (x)}} + 1$

Step 3.3.1.5.9

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{\sin (x)}{\cos (x)} = \frac{\sin^{2} (x) - {\cos (x)}^{1 + 1}}{\cos (x) \sin (x) \cdot 1 + \cos (x) \sin (x) \frac{\cos (x)}{\sin (x)}} + 1$

Step 3.3.1.5.10

Add $1$ and $1$ .

$\frac{\sin (x)}{\cos (x)} = \frac{\sin^{2} (x) - \cos^{2} (x)}{\cos (x) \sin (x) \cdot 1 + \cos (x) \sin (x) \frac{\cos (x)}{\sin (x)}} + 1$

Step 3.3.1.5.11

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

Tap for more steps...

Step 3.3.1.5.11.1

Factor $\sin (x)$ out of $\cos (x) \sin (x)$ .

$\frac{\sin (x)}{\cos (x)} = \frac{\sin^{2} (x) - \cos^{2} (x)}{\cos (x) \sin (x) \cdot 1 + \sin (x) \cos (x) \frac{\cos (x)}{\sin (x)}} + 1$

Step 3.3.1.5.11.2

Cancel the common factor.

$\frac{\sin (x)}{\cos (x)} = \frac{\sin^{2} (x) - \cos^{2} (x)}{\cos (x) \sin (x) \cdot 1 + \sin (x) \cos (x) \frac{\cos (x)}{\sin (x)}} + 1$

Step 3.3.1.5.11.3

Rewrite the expression.

$\frac{\sin (x)}{\cos (x)} = \frac{\sin^{2} (x) - \cos^{2} (x)}{\cos (x) \sin (x) \cdot 1 + \cos (x) \cos (x)} + 1$

Step 3.3.1.5.12

Raise $\cos (x)$ to the power of $1$ .

$\frac{\sin (x)}{\cos (x)} = \frac{\sin^{2} (x) - \cos^{2} (x)}{\cos (x) \sin (x) \cdot 1 + \cos^{1} (x) \cos (x)} + 1$

Step 3.3.1.5.13

Raise $\cos (x)$ to the power of $1$ .

$\frac{\sin (x)}{\cos (x)} = \frac{\sin^{2} (x) - \cos^{2} (x)}{\cos (x) \sin (x) \cdot 1 + \cos^{1} (x) \cos^{1} (x)} + 1$

Step 3.3.1.5.14

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{\sin (x)}{\cos (x)} = \frac{\sin^{2} (x) - \cos^{2} (x)}{\cos (x) \sin (x) \cdot 1 + {\cos (x)}^{1 + 1}} + 1$

Step 3.3.1.5.15

Add $1$ and $1$ .

$\frac{\sin (x)}{\cos (x)} = \frac{\sin^{2} (x) - \cos^{2} (x)}{\cos (x) \sin (x) \cdot 1 + \cos^{2} (x)} + 1$

Step 3.3.1.6

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = \sin (x)$ and $b = \cos (x)$ .

$\frac{\sin (x)}{\cos (x)} = \frac{(\sin (x) + \cos (x)) (\sin (x) - \cos (x))}{\cos (x) \sin (x) \cdot 1 + \cos^{2} (x)} + 1$

Step 3.3.1.7

Simplify the denominator.

Tap for more steps...

Step 3.3.1.7.1

Factor $\cos (x)$ out of $\cos (x) \sin (x) \cdot 1 + \cos^{2} (x)$ .

Tap for more steps...

Step 3.3.1.7.1.1

Factor $\cos (x)$ out of $\cos (x) \sin (x) \cdot 1$ .

$\frac{\sin (x)}{\cos (x)} = \frac{(\sin (x) + \cos (x)) (\sin (x) - \cos (x))}{\cos (x) (\sin (x) \cdot 1) + \cos^{2} (x)} + 1$

Step 3.3.1.7.1.2

Factor $\cos (x)$ out of $\cos^{2} (x)$ .

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

Step 3.3.1.7.1.3

Factor $\cos (x)$ out of $\cos (x) (\sin (x) \cdot 1) + \cos (x) \cos (x)$ .

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

Step 3.3.1.7.2

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

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

Step 3.3.1.8

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

Tap for more steps...

Step 3.3.1.8.1

Cancel the common factor.

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

Step 3.3.1.8.2

Rewrite the expression.

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

Step 3.4

Multiply both sides of the equation by $\cos (x)$ .

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

Step 3.5

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

Tap for more steps...

Step 3.5.1

Cancel the common factor.

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

Step 3.5.2

Rewrite the expression.

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

Step 3.6

Apply the distributive property.

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

Step 3.7

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

Tap for more steps...

Step 3.7.1

Cancel the common factor.

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

Step 3.7.2

Rewrite the expression.

$\sin (x) = \sin (x) - \cos (x) + \cos (x) \cdot 1$

Step 3.8

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

$\sin (x) = \sin (x) - \cos (x) + \cos (x)$

Step 3.9

Combine the opposite terms in $\sin (x) - \cos (x) + \cos (x)$ .

Tap for more steps...

Step 3.9.1

Add $- \cos (x)$ and $\cos (x)$ .

$\sin (x) = \sin (x) + 0$

Step 3.9.2

Add $\sin (x)$ and $0$ .

$\sin (x) = \sin (x)$

Step 3.10

For the two functions to be equal, the arguments of each must be equal.

$x = x$

Step 3.11

Move all terms containing $x$ to the left side of the equation.

Tap for more steps...

Step 3.11.1

Subtract $x$ from both sides of the equation.

$x - x = 0$

Step 3.11.2

Subtract $x$ from $x$ .

$0 = 0$

Step 3.12

Since $0 = 0$ , the equation will always be true for any value of $x$ .

All real numbers

Step 4

The result can be shown in multiple forms.

All real numbers

Interval Notation:

$(- \infty, \infty)$