Trigonometry Examples

$\cos^{2} (x) = \frac{\csc (x) \cos (x)}{\tan (x) + \cot (x)}$

Step 1

Start on the right side.

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

Step 2

Convert to sines and cosines.

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Step 2.1

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

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

Step 2.2

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

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

Step 2.3

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

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

Step 3

Simplify.

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Step 3.1

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

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

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

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

Step 3.2.2

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

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

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

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

Step 3.2.3.2

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

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

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

Step 3.2.4

Combine the numerators over the common denominator.

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

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 $1$ .

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

Step 3.2.5.1.2

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

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

Step 3.2.5.1.3

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

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

Step 3.2.5.1.4

Add $1$ and $1$ .

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

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 $1$ .

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

Step 3.2.5.2.2

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

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

Step 3.2.5.2.3

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

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

Step 3.2.5.2.4

Add $1$ and $1$ .

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

Step 3.3

Multiply the numerator by the reciprocal of the denominator.

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

Step 3.4

Cancel the common factor of $\sin (x)$ .

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Step 3.4.1

Cancel the common factor.

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

Step 3.4.2

Rewrite the expression.

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

Step 3.5

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

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

Step 3.6

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

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

Step 3.7

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

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

Step 3.8

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

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

Step 3.9

Add $1$ and $1$ .

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

Step 4

Apply pythagorean identity.

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

Step 5

Divide ${\cos (x)}^{2}$ by $1$ .

$\cos^{2} (x)$

Step 6

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

$\cos^{2} (x) = \frac{\csc (x) \cos (x)}{\tan (x) + \cot (x)}$ is an identity