Trigonometry Examples

$\frac{\cot (A)}{\csc (A) + 1} = \sec (A) - \tan (A)$

Step 1

Start on the left side.

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

Step 2

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

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

Step 3

Combine.

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

Step 4

Simplify numerator.

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Step 4.1

Apply the distributive property.

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

Step 4.2

Multiply $\cot (A)$ by $1$ .

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

Step 4.3

Reorder factors in $\cot (A) (- \csc (A)) + \cot (A)$ .

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

Step 5

Simplify denominator.

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Step 5.1

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

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Step 5.1.1

Apply the distributive property.

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

Step 5.1.2

Apply the distributive property.

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

Step 5.1.3

Apply the distributive property.

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

Step 5.2

Simplify and combine like terms.

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

Step 6

Apply Pythagorean identity.

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Step 6.1

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

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

Step 6.2

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

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

Step 6.3

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

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

Step 6.4

Apply pythagorean identity.

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

Step 7

Convert to sines and cosines.

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Step 7.1

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

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

Step 7.2

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

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

Step 7.3

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

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

Step 7.4

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

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

Step 7.5

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

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

Step 8

Simplify.

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Step 8.1

Multiply the numerator by the reciprocal of the denominator.

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

Step 8.2

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

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Step 8.2.1

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

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

Step 8.2.2

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

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

Step 8.2.3

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

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

Step 8.2.4

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

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

Step 8.2.5

Add $1$ and $1$ .

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

Step 8.3

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

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

Step 8.4

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

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Step 8.4.1

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

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

Step 8.4.2

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

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

Step 8.4.3

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

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

Step 8.4.4

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

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

Step 8.4.5

Add $1$ and $1$ .

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

Step 8.5

Combine the numerators over the common denominator.

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

Step 8.6

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

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Step 8.6.1

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

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

Step 8.6.2

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

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

Step 8.6.3

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

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

Step 8.7

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

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Step 8.7.1

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

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

Step 8.7.2

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

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

Step 8.7.3

Cancel the common factor.

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

Step 8.7.4

Rewrite the expression.

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

Step 8.8

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

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Step 8.8.1

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

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

Step 8.8.2

Cancel the common factor.

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

Step 8.8.3

Rewrite the expression.

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

Step 8.9

Move the negative in front of the fraction.

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

Step 8.10

Apply the distributive property.

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

Step 8.11

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

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Step 8.11.1

Multiply $- 1$ by $- 1$ .

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

Step 8.11.2

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

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

Step 8.12

Combine $\sin (A)$ and $\frac{1}{\cos (A)}$ .

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

Step 9

Now consider the right side of the equation.

$\sec (A) - \tan (A)$

Step 10

Convert to sines and cosines.

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Step 10.1

Apply the reciprocal identity to $\sec (A)$ .

$\frac{1}{\cos (A)} - \tan (A)$

Step 10.2

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

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

Step 11

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

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