Trigonometry Examples

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

Step 1

Start on the left side.

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

Step 2

Convert to sines and cosines.

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Step 2.1

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

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

Step 2.2

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

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

Step 2.3

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

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

Step 2.4

Apply the product rule to $\frac{\sin (x)}{\cos (x)}$ .

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

Step 3

Simplify.

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Step 3.1

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

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Step 3.1.1

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

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

Step 3.1.2

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

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

Step 3.1.3

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

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

Step 3.1.4

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

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

Step 3.1.5

Add $1$ and $1$ .

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

Step 3.2

Combine.

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

Step 3.3

Cancel the common factor of $\cos (x)$ and ${\cos (x)}^{2}$ .

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Step 3.3.1

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

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

Step 3.3.2

Cancel the common factors.

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Step 3.3.2.1

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

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

Step 3.3.2.2

Cancel the common factor.

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

Step 3.3.2.3

Rewrite the expression.

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

Step 3.4

Cancel the common factor of ${\sin (x)}^{2}$ .

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

Step 4

Rewrite $\frac{1}{\cos (x)}$ as $\sec (x)$ .

$\sec (x)$

Step 5

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

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