Trigonometry Examples

\frac{sin (x)}{1 + cos (x)} + \frac{1 + cos (x)}{sin (x)} = 2 csc (x)

$\frac{\sin (x)}{1 + \cos (x)} + \frac{1 + \cos (x)}{\sin (x)} = 2 \csc (x)$

Step 1

Start on the left side.

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

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

Step 2

Simplify the expression.

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Step 2.1

To write

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

$\frac{\sin (x)}{1 + \cos (x)}$ as a fraction with a common denominator, multiply by

\frac{sin (x)}{sin (x)}

$\frac{\sin (x)}{\sin (x)}$ .

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

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

Step 2.2

To write

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

$\frac{1 + \cos (x)}{\sin (x)}$ as a fraction with a common denominator, multiply by

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

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

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

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

Step 2.3

Write each expression with a common denominator of

(1 + cos (x)) sin (x)

$(1 + \cos (x)) \sin (x)$ , by multiplying each by an appropriate factor of

1

$1$ .

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Step 2.3.1

Multiply

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

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

\frac{sin (x)}{sin (x)}

$\frac{\sin (x)}{\sin (x)}$ .

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

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

Step 2.3.2

Multiply

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

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

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

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

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

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

Step 2.3.3

Reorder the factors of

(1 + cos (x)) sin (x)

$(1 + \cos (x)) \sin (x)$ .

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

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

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

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

Step 2.4

Combine the numerators over the common denominator.

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

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

Step 2.5

Simplify the numerator.

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Step 2.5.1

Multiply

sin (x) sin (x)

$\sin (x) \sin (x)$ .

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Step 2.5.1.1

Raise

sin (x)

$\sin (x)$ to the power of

1

$1$ .

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

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

Step 2.5.1.2

Raise

sin (x)

$\sin (x)$ to the power of

1

$1$ .

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

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

Step 2.5.1.3

Use the power rule

a^{m} a^{n} = a^{m + n}

$a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

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

Step 2.5.1.4

Add

1

$1$ and

1

$1$ .

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

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

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

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

Step 2.5.2

Expand

(1 + cos (x)) (1 + cos (x))

$(1 + \cos (x)) (1 + \cos (x))$ using the FOIL Method.

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Step 2.5.2.1

Apply the distributive property.

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

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

Step 2.5.2.2

Apply the distributive property.

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

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

Step 2.5.2.3

Apply the distributive property.

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

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

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

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

Step 2.5.3

Simplify and combine like terms.

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Step 2.5.3.1

Simplify each term.

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Step 2.5.3.1.1

Multiply

1

$1$ by

1

$1$ .

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

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

Step 2.5.3.1.2

Multiply

cos (x)

$\cos (x)$ by

1

$1$ .

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

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

Step 2.5.3.1.3

Multiply

cos (x)

$\cos (x)$ by

1

$1$ .

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

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

Step 2.5.3.1.4

Multiply

cos (x) cos (x)

$\cos (x) \cos (x)$ .

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Step 2.5.3.1.4.1

Raise

cos (x)

$\cos (x)$ to the power of

1

$1$ .

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

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

Step 2.5.3.1.4.2

Raise

cos (x)

$\cos (x)$ to the power of

1

$1$ .

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

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

Step 2.5.3.1.4.3

Use the power rule

a^{m} a^{n} = a^{m + n}

$a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

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

Step 2.5.3.1.4.4

Add

1

$1$ and

1

$1$ .

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

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

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

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

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

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

Step 2.5.3.2

Add

cos (x)

$\cos (x)$ and

cos (x)

$\cos (x)$ .

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

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

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

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

Step 2.5.4

Rewrite

{sin}^{2} (x) + 1 + 2 cos (x) + {cos}^{2} (x)

$\sin^{2} (x) + 1 + 2 \cos (x) + \cos^{2} (x)$ in a factored form.

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Step 2.5.4.1

Regroup terms.

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

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

Step 2.5.4.2

Apply pythagorean identity.

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

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

Step 2.5.4.3

Add

1

$1$ and

1

$1$ .

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

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

Step 2.5.4.4

Factor

2

$2$ out of

2 + 2 cos (x)

$2 + 2 \cos (x)$ .

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Step 2.5.4.4.1

Factor

2

$2$ out of

2

$2$ .

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

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

Step 2.5.4.4.2

Factor

2

$2$ out of

2 \cdot 1 + 2 cos (x)

$2 \cdot 1 + 2 \cos (x)$ .

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

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

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

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

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

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

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

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

Step 2.6

Cancel the common factor of

1 + cos (x)

$1 + \cos (x)$ .

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Step 2.6.1

Cancel the common factor.

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

Step 2.6.2

Rewrite the expression.

$\frac{2}{\sin (x)}$

Step 3

Rewrite

$\frac{2}{\sin (x)}$ as

$2 \csc (x)$ .

$2 \csc (x)$

Step 4

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

$\frac{\sin (x)}{1 + \cos (x)} + \frac{1 + \cos (x)}{\sin (x)} = 2 \csc (x)$ is an identity