Calculus Examples

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

Step 1

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

$(\sec (2 x) + \csc (2 x)) (\sec (2 x) + \csc (2 x))$

Step 2

Expand $(\sec (2 x) + \csc (2 x)) (\sec (2 x) + \csc (2 x))$ using the FOIL Method.

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Step 2.1

Apply the distributive property.

$\sec (2 x) (\sec (2 x) + \csc (2 x)) + \csc (2 x) (\sec (2 x) + \csc (2 x))$

Step 2.2

Apply the distributive property.

$\sec (2 x) \sec (2 x) + \sec (2 x) \csc (2 x) + \csc (2 x) (\sec (2 x) + \csc (2 x))$

Step 2.3

Apply the distributive property.

$\sec (2 x) \sec (2 x) + \sec (2 x) \csc (2 x) + \csc (2 x) \sec (2 x) + \csc (2 x) \csc (2 x)$

Step 3

Simplify and combine like terms.

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Step 3.1

Simplify each term.

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Step 3.1.1

Multiply $\sec (2 x) \sec (2 x)$ .

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Step 3.1.1.1

Raise $\sec (2 x)$ to the power of $1$ .

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

Step 3.1.1.2

Raise $\sec (2 x)$ to the power of $1$ .

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

Step 3.1.1.3

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

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

Step 3.1.1.4

Add $1$ and $1$ .

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

Step 3.1.2

Multiply $\csc (2 x) \csc (2 x)$ .

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Step 3.1.2.1

Raise $\csc (2 x)$ to the power of $1$ .

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

Step 3.1.2.2

Raise $\csc (2 x)$ to the power of $1$ .

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

Step 3.1.2.3

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

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

Step 3.1.2.4

Add $1$ and $1$ .

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

Step 3.2

Reorder the factors of $\sec (2 x) \csc (2 x)$ .

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

Step 3.3

Add $\csc (2 x) \sec (2 x)$ and $\csc (2 x) \sec (2 x)$ .

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