Trigonometry Examples

$\frac{1 + \cos (3 t)}{\sin (3 t)} + \frac{\sin (3 t)}{1 + \cos (3 t)} = 2 \csc (3 t)$

Step 1

Start on the left side.

$\frac{1 + \cos (3 t)}{\sin (3 t)} + \frac{\sin (3 t)}{1 + \cos (3 t)}$

Step 2

Simplify the expression.

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Step 2.1

To write $\frac{1 + \cos (3 t)}{\sin (3 t)}$ as a fraction with a common denominator, multiply by $\frac{1 + \cos (3 t)}{1 + \cos (3 t)}$ .

$\frac{1 + \cos (3 t)}{\sin (3 t)} \cdot \frac{1 + \cos (3 t)}{1 + \cos (3 t)} + \frac{\sin (3 t)}{1 + \cos (3 t)}$

Step 2.2

To write $\frac{\sin (3 t)}{1 + \cos (3 t)}$ as a fraction with a common denominator, multiply by $\frac{\sin (3 t)}{\sin (3 t)}$ .

$\frac{1 + \cos (3 t)}{\sin (3 t)} \cdot \frac{1 + \cos (3 t)}{1 + \cos (3 t)} + \frac{\sin (3 t)}{1 + \cos (3 t)} \cdot \frac{\sin (3 t)}{\sin (3 t)}$

Step 2.3

Write each expression with a common denominator of $\sin (3 t) (1 + \cos (3 t))$ , by multiplying each by an appropriate factor of $1$ .

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Step 2.3.1

Multiply $\frac{1 + \cos (3 t)}{\sin (3 t)}$ by $\frac{1 + \cos (3 t)}{1 + \cos (3 t)}$ .

$\frac{(1 + \cos (3 t)) (1 + \cos (3 t))}{\sin (3 t) (1 + \cos (3 t))} + \frac{\sin (3 t)}{1 + \cos (3 t)} \cdot \frac{\sin (3 t)}{\sin (3 t)}$

Step 2.3.2

Multiply $\frac{\sin (3 t)}{1 + \cos (3 t)}$ by $\frac{\sin (3 t)}{\sin (3 t)}$ .

$\frac{(1 + \cos (3 t)) (1 + \cos (3 t))}{\sin (3 t) (1 + \cos (3 t))} + \frac{\sin (3 t) \sin (3 t)}{(1 + \cos (3 t)) \sin (3 t)}$

Step 2.3.3

Reorder the factors of $(1 + \cos (3 t)) \sin (3 t)$ .

$\frac{(1 + \cos (3 t)) (1 + \cos (3 t))}{\sin (3 t) (1 + \cos (3 t))} + \frac{\sin (3 t) \sin (3 t)}{\sin (3 t) (1 + \cos (3 t))}$

Step 2.4

Combine the numerators over the common denominator.

$\frac{(1 + \cos (3 t)) (1 + \cos (3 t)) + \sin (3 t) \sin (3 t)}{\sin (3 t) (1 + \cos (3 t))}$

Step 2.5

Simplify the numerator.

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Step 2.5.1

Expand $(1 + \cos (3 t)) (1 + \cos (3 t))$ using the FOIL Method.

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Step 2.5.1.1

Apply the distributive property.

$\frac{1 (1 + \cos (3 t)) + \cos (3 t) (1 + \cos (3 t)) + \sin (3 t) \sin (3 t)}{\sin (3 t) (1 + \cos (3 t))}$

Step 2.5.1.2

Apply the distributive property.

$\frac{1 \cdot 1 + 1 \cos (3 t) + \cos (3 t) (1 + \cos (3 t)) + \sin (3 t) \sin (3 t)}{\sin (3 t) (1 + \cos (3 t))}$

Step 2.5.1.3

Apply the distributive property.

$\frac{1 \cdot 1 + 1 \cos (3 t) + \cos (3 t) \cdot 1 + \cos (3 t) \cos (3 t) + \sin (3 t) \sin (3 t)}{\sin (3 t) (1 + \cos (3 t))}$

Step 2.5.2

Simplify and combine like terms.

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Step 2.5.2.1

Simplify each term.

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Step 2.5.2.1.1

Multiply $1$ by $1$ .

$\frac{1 + 1 \cos (3 t) + \cos (3 t) \cdot 1 + \cos (3 t) \cos (3 t) + \sin (3 t) \sin (3 t)}{\sin (3 t) (1 + \cos (3 t))}$

Step 2.5.2.1.2

Multiply $\cos (3 t)$ by $1$ .

$\frac{1 + \cos (3 t) + \cos (3 t) \cdot 1 + \cos (3 t) \cos (3 t) + \sin (3 t) \sin (3 t)}{\sin (3 t) (1 + \cos (3 t))}$

Step 2.5.2.1.3

Multiply $\cos (3 t)$ by $1$ .

$\frac{1 + \cos (3 t) + \cos (3 t) + \cos (3 t) \cos (3 t) + \sin (3 t) \sin (3 t)}{\sin (3 t) (1 + \cos (3 t))}$

Step 2.5.2.1.4

Multiply $\cos (3 t) \cos (3 t)$ .

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Step 2.5.2.1.4.1

Raise $\cos (3 t)$ to the power of $1$ .

$\frac{1 + \cos (3 t) + \cos (3 t) + \cos^{1} (3 t) \cos (3 t) + \sin (3 t) \sin (3 t)}{\sin (3 t) (1 + \cos (3 t))}$

Step 2.5.2.1.4.2

Raise $\cos (3 t)$ to the power of $1$ .

$\frac{1 + \cos (3 t) + \cos (3 t) + \cos^{1} (3 t) \cos^{1} (3 t) + \sin (3 t) \sin (3 t)}{\sin (3 t) (1 + \cos (3 t))}$

Step 2.5.2.1.4.3

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

$\frac{1 + \cos (3 t) + \cos (3 t) + {\cos (3 t)}^{1 + 1} + \sin (3 t) \sin (3 t)}{\sin (3 t) (1 + \cos (3 t))}$

Step 2.5.2.1.4.4

Add $1$ and $1$ .

$\frac{1 + \cos (3 t) + \cos (3 t) + \cos^{2} (3 t) + \sin (3 t) \sin (3 t)}{\sin (3 t) (1 + \cos (3 t))}$

Step 2.5.2.2

Add $\cos (3 t)$ and $\cos (3 t)$ .

$\frac{1 + 2 \cos (3 t) + \cos^{2} (3 t) + \sin (3 t) \sin (3 t)}{\sin (3 t) (1 + \cos (3 t))}$

Step 2.5.3

Multiply $\sin (3 t) \sin (3 t)$ .

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Step 2.5.3.1

Raise $\sin (3 t)$ to the power of $1$ .

$\frac{1 + 2 \cos (3 t) + \cos^{2} (3 t) + \sin^{1} (3 t) \sin (3 t)}{\sin (3 t) (1 + \cos (3 t))}$

Step 2.5.3.2

Raise $\sin (3 t)$ to the power of $1$ .

$\frac{1 + 2 \cos (3 t) + \cos^{2} (3 t) + \sin^{1} (3 t) \sin^{1} (3 t)}{\sin (3 t) (1 + \cos (3 t))}$

Step 2.5.3.3

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

$\frac{1 + 2 \cos (3 t) + \cos^{2} (3 t) + {\sin (3 t)}^{1 + 1}}{\sin (3 t) (1 + \cos (3 t))}$

Step 2.5.3.4

Add $1$ and $1$ .

$\frac{1 + 2 \cos (3 t) + \cos^{2} (3 t) + \sin^{2} (3 t)}{\sin (3 t) (1 + \cos (3 t))}$

Step 2.5.4

Rewrite $1 + 2 \cos (3 t) + \cos^{2} (3 t) + \sin^{2} (3 t)$ in a factored form.

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Step 2.5.4.1

Rearrange terms.

$\frac{1 + 2 \cos (3 t) + \sin^{2} (3 t) + \cos^{2} (3 t)}{\sin (3 t) (1 + \cos (3 t))}$

Step 2.5.4.2

Apply pythagorean identity.

$\frac{1 + 2 \cos (3 t) + 1}{\sin (3 t) (1 + \cos (3 t))}$

Step 2.5.4.3

Add $1$ and $1$ .

$\frac{2 + 2 \cos (3 t)}{\sin (3 t) (1 + \cos (3 t))}$

Step 2.5.4.4

Factor $2$ out of $2 + 2 \cos (3 t)$ .

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Step 2.5.4.4.1

Factor $2$ out of $2$ .

$\frac{2 \cdot 1 + 2 \cos (3 t)}{\sin (3 t) (1 + \cos (3 t))}$

Step 2.5.4.4.2

Factor $2$ out of $2 \cdot 1 + 2 \cos (3 t)$ .

$\frac{2 (1 + \cos (3 t))}{\sin (3 t) (1 + \cos (3 t))}$

Step 2.6

Cancel the common factor of $1 + \cos (3 t)$ .

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Step 2.6.1

Cancel the common factor.

$\frac{2 (1 + \cos (3 t))}{\sin (3 t) (1 + \cos (3 t))}$

Step 2.6.2

Rewrite the expression.

$\frac{2}{\sin (3 t)}$

Step 3

Rewrite $\frac{2}{\sin (3 t)}$ as $2 \csc (3 t)$ .

$2 \csc (3 t)$

Step 4

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

$\frac{1 + \cos (3 t)}{\sin (3 t)} + \frac{\sin (3 t)}{1 + \cos (3 t)} = 2 \csc (3 t)$ is an identity