Trigonometry Examples

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

Step 1

Start on the left side.

$\frac{1}{\tan (x) \csc (x) \sin (x)}$

Step 2

Convert to sines and cosines.

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Step 2.1

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

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

Step 2.2

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

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

Step 3

Simplify.

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Step 3.1

Simplify the denominator.

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Step 3.1.1

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

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

Step 3.1.2

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

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

Step 3.2

Simplify the numerator.

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Step 3.2.1

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

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

Step 3.2.2

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

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

Step 3.2.3

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

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

Step 3.2.4

Add $1$ and $1$ .

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

Step 3.3

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

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Step 3.3.1

Factor $\sin (x)$ out of ${\sin (x)}^{2}$ .

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

Step 3.3.2

Cancel the common factor.

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

Step 3.3.3

Rewrite the expression.

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

Step 3.4

Multiply the numerator by the reciprocal of the denominator.

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

Step 3.5

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

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

Step 4

Rewrite $\frac{\cos (x)}{\sin (x)}$ as $\cot (x)$ .

$\cot (x)$

Step 5

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

$\frac{1}{\tan (x) \csc (x) \sin (x)} = \cot (x)$ is an identity