Precalculus Examples

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

Step 1

Start on the left side.

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

Step 2

Convert to sines and cosines.

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Step 2.1

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

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

Step 2.2

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

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

Step 2.3

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

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

Step 3

Simplify.

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Step 3.1

Combine the numerators over the common denominator.

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

Step 3.2

Add $- \cos (x)$ and $\cos (x)$ .

$\frac{1}{\sin (x)} (\frac{1}{\sin (x)} - \sin (x)) + \frac{\sin (x) + 0}{\sin (x)}$

Step 3.3

Add $\sin (x)$ and $0$ .

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

Step 3.4

Simplify each term.

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Step 3.4.1

Apply the distributive property.

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

Step 3.4.2

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

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Step 3.4.2.1

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

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

Step 3.4.2.2

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

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

Step 3.4.2.3

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

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

Step 3.4.2.4

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

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

Step 3.4.2.5

Add $1$ and $1$ .

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

Step 3.4.3

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

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Step 3.4.3.1

Factor $\sin (x)$ out of $- \sin (x)$ .

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

Step 3.4.3.2

Cancel the common factor.

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

Step 3.4.3.3

Rewrite the expression.

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

Step 3.4.4

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

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Step 3.4.4.1

Cancel the common factor.

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

Step 3.4.4.2

Rewrite the expression.

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

Step 3.5

Add $- 1$ and $1$ .

$\frac{1}{{\sin (x)}^{2}} + 0$

Step 3.6

Add $\frac{1}{{\sin (x)}^{2}}$ and $0$ .

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

Step 4

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

$\csc^{2} (x)$

Step 5

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

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