Trigonometry Examples

${(\cos (t) - \sec (t))}^{2}$

Step 1

Rewrite $\sec (t)$ in terms of sines and cosines.

${(\cos (t) - \frac{1}{\cos (t)})}^{2}$

Step 2

Rewrite ${(\cos (t) - \frac{1}{\cos (t)})}^{2}$ as $(\cos (t) - \frac{1}{\cos (t)}) (\cos (t) - \frac{1}{\cos (t)})$ .

$(\cos (t) - \frac{1}{\cos (t)}) (\cos (t) - \frac{1}{\cos (t)})$

Step 3

Expand $(\cos (t) - \frac{1}{\cos (t)}) (\cos (t) - \frac{1}{\cos (t)})$ using the FOIL Method.

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Step 3.1

Apply the distributive property.

$\cos (t) (\cos (t) - \frac{1}{\cos (t)}) - \frac{1}{\cos (t)} (\cos (t) - \frac{1}{\cos (t)})$

Step 3.2

Apply the distributive property.

$\cos (t) \cos (t) + \cos (t) (- \frac{1}{\cos (t)}) - \frac{1}{\cos (t)} (\cos (t) - \frac{1}{\cos (t)})$

Step 3.3

Apply the distributive property.

$\cos (t) \cos (t) + \cos (t) (- \frac{1}{\cos (t)}) - \frac{1}{\cos (t)} \cos (t) - \frac{1}{\cos (t)} (- \frac{1}{\cos (t)})$

Step 4

Simplify and combine like terms.

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Step 4.1

Simplify each term.

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Step 4.1.1

Multiply $\cos (t) \cos (t)$ .

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Step 4.1.1.1

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

$\cos^{1} (t) \cos (t) + \cos (t) (- \frac{1}{\cos (t)}) - \frac{1}{\cos (t)} \cos (t) - \frac{1}{\cos (t)} (- \frac{1}{\cos (t)})$

Step 4.1.1.2

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

$\cos^{1} (t) \cos^{1} (t) + \cos (t) (- \frac{1}{\cos (t)}) - \frac{1}{\cos (t)} \cos (t) - \frac{1}{\cos (t)} (- \frac{1}{\cos (t)})$

Step 4.1.1.3

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

${\cos (t)}^{1 + 1} + \cos (t) (- \frac{1}{\cos (t)}) - \frac{1}{\cos (t)} \cos (t) - \frac{1}{\cos (t)} (- \frac{1}{\cos (t)})$

Step 4.1.1.4

Add $1$ and $1$ .

$\cos^{2} (t) + \cos (t) (- \frac{1}{\cos (t)}) - \frac{1}{\cos (t)} \cos (t) - \frac{1}{\cos (t)} (- \frac{1}{\cos (t)})$

Step 4.1.2

Rewrite using the commutative property of multiplication.

$\cos^{2} (t) - \cos (t) \frac{1}{\cos (t)} - \frac{1}{\cos (t)} \cos (t) - \frac{1}{\cos (t)} (- \frac{1}{\cos (t)})$

Step 4.1.3

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

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Step 4.1.3.1

Factor $\cos (t)$ out of $- \cos (t)$ .

$\cos^{2} (t) + \cos (t) \cdot - 1 \frac{1}{\cos (t)} - \frac{1}{\cos (t)} \cos (t) - \frac{1}{\cos (t)} (- \frac{1}{\cos (t)})$

Step 4.1.3.2

Cancel the common factor.

$\cos^{2} (t) + \cos (t) \cdot - 1 \frac{1}{\cos (t)} - \frac{1}{\cos (t)} \cos (t) - \frac{1}{\cos (t)} (- \frac{1}{\cos (t)})$

Step 4.1.3.3

Rewrite the expression.

$\cos^{2} (t) - 1 - \frac{1}{\cos (t)} \cos (t) - \frac{1}{\cos (t)} (- \frac{1}{\cos (t)})$

Step 4.1.4

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

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Step 4.1.4.1

Move the leading negative in $- \frac{1}{\cos (t)}$ into the numerator.

$\cos^{2} (t) - 1 + \frac{- 1}{\cos (t)} \cos (t) - \frac{1}{\cos (t)} (- \frac{1}{\cos (t)})$

Step 4.1.4.2

Cancel the common factor.

$\cos^{2} (t) - 1 + \frac{- 1}{\cos (t)} \cos (t) - \frac{1}{\cos (t)} (- \frac{1}{\cos (t)})$

Step 4.1.4.3

Rewrite the expression.

$\cos^{2} (t) - 1 - 1 - \frac{1}{\cos (t)} (- \frac{1}{\cos (t)})$

Step 4.1.5

Multiply $- \frac{1}{\cos (t)} (- \frac{1}{\cos (t)})$ .

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Step 4.1.5.1

Multiply $- 1$ by $- 1$ .

$\cos^{2} (t) - 1 - 1 + 1 \frac{1}{\cos (t)} \frac{1}{\cos (t)}$

Step 4.1.5.2

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

$\cos^{2} (t) - 1 - 1 + \frac{1}{\cos (t)} \cdot \frac{1}{\cos (t)}$

Step 4.1.5.3

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

$\cos^{2} (t) - 1 - 1 + \frac{1}{\cos (t) \cos (t)}$

Step 4.1.5.4

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

$\cos^{2} (t) - 1 - 1 + \frac{1}{\cos^{1} (t) \cos (t)}$

Step 4.1.5.5

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

$\cos^{2} (t) - 1 - 1 + \frac{1}{\cos^{1} (t) \cos^{1} (t)}$

Step 4.1.5.6

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

$\cos^{2} (t) - 1 - 1 + \frac{1}{{\cos (t)}^{1 + 1}}$

Step 4.1.5.7

Add $1$ and $1$ .

$\cos^{2} (t) - 1 - 1 + \frac{1}{\cos^{2} (t)}$

Step 4.2

Subtract $1$ from $- 1$ .

$\cos^{2} (t) - 2 + \frac{1}{\cos^{2} (t)}$

Step 5

Simplify each term.

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Step 5.1

Rewrite $1$ as $1^{2}$ .

$\cos^{2} (t) - 2 + \frac{1^{2}}{\cos^{2} (t)}$

Step 5.2

Rewrite $\frac{1^{2}}{\cos^{2} (t)}$ as ${(\frac{1}{\cos (t)})}^{2}$ .

$\cos^{2} (t) - 2 + {(\frac{1}{\cos (t)})}^{2}$

Step 5.3

Convert from $\frac{1}{\cos (t)}$ to $\sec (t)$ .

$\cos^{2} (t) - 2 + \sec^{2} (t)$