Trigonometry Examples

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

Step 1

Start on the left side.

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

Step 2

Convert to sines and cosines.

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Step 2.1

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

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

Step 2.2

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

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

Step 2.3

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

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

Step 3

Simplify.

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Step 3.1

Multiply the numerator by the reciprocal of the denominator.

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

Step 3.2

Simplify the denominator.

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Step 3.2.1

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

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

Step 3.2.2

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

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

Step 3.2.3

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

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Step 3.2.3.1

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

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

Step 3.2.3.2

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

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

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

Step 3.2.4

Combine the numerators over the common denominator.

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

Step 3.2.5

Simplify the numerator.

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Step 3.2.5.1

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

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Step 3.2.5.1.1

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

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

Step 3.2.5.1.2

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

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

Step 3.2.5.1.3

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

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

Step 3.2.5.1.4

Add $1$ and $1$ .

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

Step 3.2.5.2

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

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Step 3.2.5.2.1

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

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

Step 3.2.5.2.2

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

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

Step 3.2.5.2.3

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

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

Step 3.2.5.2.4

Add $1$ and $1$ .

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

Step 3.3

Combine.

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

Step 3.4

Multiply $1$ by $1$ .

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

Step 3.5

Combine $\sin (x)$ and $\frac{{\sin (x)}^{2} + {\cos (x)}^{2}}{\cos (x) \sin (x)}$ .

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

Step 3.6

Reduce the expression $\frac{\sin (x) ({\sin (x)}^{2} + {\cos (x)}^{2})}{\cos (x) \sin (x)}$ by cancelling the common factors.

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Step 3.6.1

Cancel the common factor.

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

Step 3.6.2

Rewrite the expression.

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

Step 3.7

Multiply the numerator by the reciprocal of the denominator.

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

Step 3.8

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

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

Step 4

Apply pythagorean identity.

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

Step 5

Divide $\cos (x)$ by $1$ .

$\cos (x)$

Step 6

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

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