Trigonometry Examples

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

Step 1

Start on the left side.

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

Step 2

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

$\frac{\cot (x)}{\csc (x) - 1} \cdot \frac{- \csc (x) - 1}{- \csc (x) - 1}$

Step 3

Combine.

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

Step 4

Simplify numerator.

Tap for more steps...

Apply the distributive property.

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

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

$\frac{\cot (x) (- \csc (x)) - 1 \cdot \cot (x)}{(\csc (x) - 1) (- \csc (x) - 1)}$

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

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

Reorder factors in $\cot (x) (- \csc (x)) - \cot (x)$ .

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

Step 5

Simplify denominator.

Tap for more steps...

Expand $(\csc (x) - 1) (- \csc (x) - 1)$ using the FOIL Method.

Tap for more steps...

Apply the distributive property.

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

Apply the distributive property.

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

Apply the distributive property.

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

Simplify and combine like terms.

$\frac{- \cot (x) \csc (x) - \cot (x)}{- \csc^{2} (x) + 1}$

Step 6

Apply Pythagorean identity.

Tap for more steps...

Factor $- 1$ out of $- \csc^{2} (x)$ .

$\frac{- \cot (x) \csc (x) - \cot (x)}{- (\csc^{2} (x)) + 1}$

Rewrite $1$ as $- 1 (- 1)$ .

$\frac{- \cot (x) \csc (x) - \cot (x)}{- \csc^{2} (x) - 1 \cdot - 1}$

Factor $- 1$ out of $- \csc^{2} (x) - 1 \cdot - 1$ .

$\frac{- \cot (x) \csc (x) - \cot (x)}{- (\csc^{2} (x) - 1)}$

Apply pythagorean identity.

$\frac{- \cot (x) \csc (x) - \cot (x)}{- \cot^{2} (x)}$

Step 7

Convert to sines and cosines.

Tap for more steps...

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

$\frac{- \frac{\cos (x)}{\sin (x)} \csc (x) - \cot (x)}{- \cot^{2} (x)}$

Apply the reciprocal identity to $\csc (x)$ .

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

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

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

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

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

Apply the product rule to $\frac{\cos (x)}{\sin (x)}$ .

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

Step 8

Simplify.

Tap for more steps...

Multiply the numerator by the reciprocal of the denominator.

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

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

Tap for more steps...

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

$(- \frac{\cos (x)}{\sin (x) \sin (x)} - \frac{\cos (x)}{\sin (x)}) (- \frac{{\sin (x)}^{2}}{{\cos (x)}^{2}})$

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

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

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

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

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

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

Add $1$ and $1$ .

$(- \frac{\cos (x)}{{\sin (x)}^{2}} - \frac{\cos (x)}{\sin (x)}) (- \frac{{\sin (x)}^{2}}{{\cos (x)}^{2}})$

To write $- \frac{\cos (x)}{\sin (x)}$ as a fraction with a common denominator, multiply by $\frac{\sin (x)}{\sin (x)}$ .

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

Write each expression with a common denominator of ${\sin (x)}^{2}$ , by multiplying each by an appropriate factor of $1$ .

Tap for more steps...

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

$(- \frac{\cos (x)}{{\sin (x)}^{2}} - \frac{\cos (x) \sin (x)}{\sin (x) \sin (x)}) (- \frac{{\sin (x)}^{2}}{{\cos (x)}^{2}})$

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

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

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

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

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

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

Add $1$ and $1$ .

$(- \frac{\cos (x)}{{\sin (x)}^{2}} - \frac{\cos (x) \sin (x)}{{\sin (x)}^{2}}) (- \frac{{\sin (x)}^{2}}{{\cos (x)}^{2}})$

Combine the numerators over the common denominator.

$\frac{- \cos (x) - \cos (x) \sin (x)}{{\sin (x)}^{2}} (- \frac{{\sin (x)}^{2}}{{\cos (x)}^{2}})$

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

Tap for more steps...

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

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

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

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

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

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

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

Tap for more steps...

Move the leading negative in $- \frac{{\sin (x)}^{2}}{{\cos (x)}^{2}}$ into the numerator.

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

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

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

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

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

Cancel the common factor.

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

Rewrite the expression.

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

Cancel the common factor of ${\sin (x)}^{2}$ .

Tap for more steps...

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

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

Cancel the common factor.

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

Rewrite the expression.

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

Apply the distributive property.

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

Multiply $- 1$ by $1$ .

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

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

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

Move the negative in front of the fraction.

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

Apply the distributive property.

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

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

Tap for more steps...

Multiply $- 1$ by $- 1$ .

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

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

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

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

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

Step 9

Now consider the right side of the equation.

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

Step 10

Convert to sines and cosines.

Tap for more steps...

Apply the reciprocal identity to $\csc (x)$ .

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

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

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

Step 11

Simplify.

Tap for more steps...

Multiply the numerator by the reciprocal of the denominator.

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

Apply the distributive property.

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

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

Tap for more steps...

Cancel the common factor.

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

Rewrite the expression.

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

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

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

Step 12

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

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