Trigonometry Examples

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

Step 1

Start on the left side.

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

Step 2

Simplify the expression.

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Step 2.1

Simplify the numerator.

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Step 2.1.1

Rewrite $\tan (x)$ in terms of sines and cosines.

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

Step 2.1.2

Rewrite $\cot (x)$ in terms of sines and cosines.

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

Step 2.2

Rewrite $\tan (x)$ in terms of sines and cosines.

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

Step 2.3

Multiply the numerator by the reciprocal of the denominator.

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

Step 2.4

Apply the distributive property.

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

Step 2.5

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

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Step 2.5.1

Cancel the common factor.

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

Step 2.5.2

Rewrite the expression.

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

Step 2.6

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

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Step 2.6.1

Cancel the common factor.

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

Step 2.6.2

Rewrite the expression.

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

Step 2.7

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

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Step 2.7.1

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

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

Step 2.7.2

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

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

Step 2.7.3

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

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

Step 2.7.4

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

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

Step 2.7.5

Add $1$ and $1$ .

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

Step 2.7.6

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

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

Step 2.7.7

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

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

Step 2.7.8

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

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

Step 2.7.9

Add $1$ and $1$ .

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

Step 3

Apply Pythagorean identity in reverse.

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

Step 4

Simplify.

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Step 4.1

Simplify the numerator.

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Step 4.1.1

Rewrite $1$ as $1^{2}$ .

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

Step 4.1.2

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = 1$ and $b = \sin (x)$ .

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

Step 4.2

Write $1$ as a fraction with a common denominator.

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

Step 4.3

Combine the numerators over the common denominator.

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

Step 4.4

Simplify the numerator.

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

Step 5

Rewrite $\frac{1}{\sin^{2} (x)}$ as $\csc^{2} (x)$ .

$\csc^{2} (x)$

Step 6

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

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