Trigonometry Examples

\frac{\cot^{2} (x)}{\csc (x)} = \csc (x) - \sin (x)

$\frac{\cot^{2} (x)}{\csc (x)} = \csc (x) - \sin (x)$

Step 1

Start on the left side.

\frac{\cot^{2} (x)}{\csc (x)}

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

Step 2

Apply Pythagorean identity in reverse.

\frac{\csc^{2} (x) - 1}{\csc (x)}

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

Step 3

Convert to sines and cosines.

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Step 3.1

Apply the reciprocal identity to

\csc (x)

$\csc (x)$ .

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

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

Step 3.2

Apply the reciprocal identity to

\csc (x)

$\csc (x)$ .

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

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

Step 3.3

Apply the product rule to

\frac{1}{\sin (x)}

$\frac{1}{\sin (x)}$ .

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

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

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

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

Step 4

Simplify.

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Step 4.1

Multiply the numerator by the reciprocal of the denominator.

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

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

Step 4.2

One to any power is one.

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

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

Step 4.3

Apply the distributive property.

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

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

Step 4.4

Cancel the common factor of

\sin (x)

$\sin (x)$ .

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Step 4.4.1

Factor

\sin (x)

$\sin (x)$ out of

{\sin (x)}^{2}

${\sin (x)}^{2}$ .

\frac{1}{\sin (x) \sin (x)} \sin (x) - 1 \sin (x)

$\frac{1}{\sin (x) \sin (x)} \sin (x) - 1 \sin (x)$

Step 4.4.2

Cancel the common factor.

\frac{1}{\sin (x) \sin (x)} \sin (x) - 1 \sin (x)

$\frac{1}{\sin (x) \sin (x)} \sin (x) - 1 \sin (x)$

Step 4.4.3

Rewrite the expression.

\frac{1}{\sin (x)} - 1 \sin (x)

$\frac{1}{\sin (x)} - 1 \sin (x)$

\frac{1}{\sin (x)} - 1 \sin (x)

$\frac{1}{\sin (x)} - 1 \sin (x)$

Step 4.5

Rewrite

- 1 \sin (x)

$- 1 \sin (x)$ as

- \sin (x)

$- \sin (x)$ .

\frac{1}{\sin (x)} - \sin (x)

$\frac{1}{\sin (x)} - \sin (x)$

\frac{1}{\sin (x)} - \sin (x)

$\frac{1}{\sin (x)} - \sin (x)$

Step 5

Now consider the right side of the equation.

\csc (x) - \sin (x)

$\csc (x) - \sin (x)$

Step 6

Apply the reciprocal identity to

\csc (x)

$\csc (x)$ .

\frac{1}{\sin (x)} - \sin (x)

$\frac{1}{\sin (x)} - \sin (x)$

Step 7

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

\frac{\cot^{2} (x)}{\csc (x)} = \csc (x) - \sin (x)

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