Precalculus Examples

$\frac{1}{\tan (θ) + \cot (θ)} = \frac{\cos (θ)}{\csc (θ)}$

Step 1

Start on the left side.

$\frac{1}{\tan (θ) + \cot (θ)}$

Step 2

Convert to sines and cosines.

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Step 2.1

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

$\frac{1}{\frac{\sin (θ)}{\cos (θ)} + \cot (θ)}$

Step 2.2

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

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

Step 3

Simplify.

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Step 3.1

Simplify the denominator.

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Step 3.1.1

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

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

Step 3.1.2

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

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

Step 3.1.3

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

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Step 3.1.3.1

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

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

Step 3.1.3.2

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

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

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

Step 3.1.4

Combine the numerators over the common denominator.

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

Step 3.1.5

Simplify the numerator.

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Step 3.1.5.1

Multiply $\sin (θ) \sin (θ)$ .

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Step 3.1.5.1.1

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

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

Step 3.1.5.1.2

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

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

Step 3.1.5.1.3

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

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

Step 3.1.5.1.4

Add $1$ and $1$ .

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

Step 3.1.5.2

Multiply $\cos (θ) \cos (θ)$ .

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Step 3.1.5.2.1

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

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

Step 3.1.5.2.2

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

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

Step 3.1.5.2.3

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

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

Step 3.1.5.2.4

Add $1$ and $1$ .

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

Step 3.2

Multiply the numerator by the reciprocal of the denominator.

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

Step 3.3

Multiply $\frac{\cos (θ) \sin (θ)}{{\sin (θ)}^{2} + {\cos (θ)}^{2}}$ by $1$ .

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

Step 4

Apply pythagorean identity.

$\frac{\cos (θ) \sin (θ)}{1}$

Step 5

Divide $\cos (θ) \sin (θ)$ by $1$ .

$\cos (θ) \sin (θ)$

Step 6

Rewrite $\cos (θ) \sin (θ)$ as $\frac{\cos (θ)}{\csc (θ)}$ .

$\frac{\cos (θ)}{\csc (θ)}$

Step 7

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

$\frac{1}{\tan (θ) + \cot (θ)} = \frac{\cos (θ)}{\csc (θ)}$ is an identity