Algebra Examples

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

Step 1

Start on the left side.

$\sec (x) \csc (x) (\tan (x) + \cot (x))$

Step 2

Convert to sines and cosines.

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Step 2.1

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

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

Step 2.2

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

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

Step 2.3

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

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

Step 2.4

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

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

Step 3

Simplify.

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Step 3.1

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

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

Step 3.2

Apply the distributive property.

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

Step 3.3

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

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Step 3.3.1

Cancel the common factor.

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

Step 3.3.2

Rewrite the expression.

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

Step 3.4

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

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

Step 3.5

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

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

Step 3.6

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

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

Step 3.7

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

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

Step 3.8

Add $1$ and $1$ .

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

Step 3.9

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

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Step 3.9.1

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

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

Step 3.9.2

Cancel the common factor.

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

Step 3.9.3

Rewrite the expression.

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

Step 3.10

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

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

Step 3.11

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

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

Step 3.12

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

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

Step 3.13

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

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

Step 3.14

Add $1$ and $1$ .

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

Step 4

Now consider the right side of the equation.

$\sec^{2} (x) + \csc^{2} (x)$

Step 5

Convert to sines and cosines.

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Step 5.1

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

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

Step 5.2

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

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

Step 5.3

Apply the product rule to $\frac{1}{\cos (x)}$ .

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

Step 5.4

Apply the product rule to $\frac{1}{\sin (x)}$ .

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

Step 6

Simplify each term.

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

Step 7

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

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