Algebra Examples

$[\begin{array}{c} 1 & 0 & \sin (x) \\ 1 & 0 & \cos (x) \\ 1 & \sec (x) & - \sin (x) \end{array}]$

Step 1

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

$[\begin{array}{c} 1 & 0 & \sin (x) \\ 1 & 0 & \cos (x) \\ 1 & \frac{1}{\cos (x)} & - \sin (x) \end{array}]$

Step 2

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

$[\begin{array}{c} 1 & 0 & \sin (x) \\ 1 & 0 & \cos (x) \\ 1 & \sec (x) & - \sin (x) \end{array}]$

Step 3

Choose the row or column with the most $0$ elements. If there are no $0$ elements choose any row or column. Multiply every element in column $2$ by its cofactor and add.

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Step 3.1

Consider the corresponding sign chart.

$| \begin{array}{c} + & - & + \\ - & + & - \\ + & - & + \end{array} |$

Step 3.2

The cofactor is the minor with the sign changed if the indices match a $-$ position on the sign chart.

Step 3.3

The minor for $a_{12}$ is the determinant with row $1$ and column $2$ deleted.

$| \begin{array}{c} 1 & \cos (x) \\ 1 & - \sin (x) \end{array} |$

Step 3.4

Multiply element $a_{12}$ by its cofactor.

$0 | \begin{array}{c} 1 & \cos (x) \\ 1 & - \sin (x) \end{array} |$

Step 3.5

The minor for $a_{22}$ is the determinant with row $2$ and column $2$ deleted.

$| \begin{array}{c} 1 & \sin (x) \\ 1 & - \sin (x) \end{array} |$

Step 3.6

Multiply element $a_{22}$ by its cofactor.

$0 | \begin{array}{c} 1 & \sin (x) \\ 1 & - \sin (x) \end{array} |$

Step 3.7

The minor for $a_{32}$ is the determinant with row $3$ and column $2$ deleted.

$| \begin{array}{c} 1 & \sin (x) \\ 1 & \cos (x) \end{array} |$

Step 3.8

Multiply element $a_{32}$ by its cofactor.

$- \sec (x) | \begin{array}{c} 1 & \sin (x) \\ 1 & \cos (x) \end{array} |$

Step 3.9

Add the terms together.

$0 | \begin{array}{c} 1 & \cos (x) \\ 1 & - \sin (x) \end{array} | + 0 | \begin{array}{c} 1 & \sin (x) \\ 1 & - \sin (x) \end{array} | - \sec (x) | \begin{array}{c} 1 & \sin (x) \\ 1 & \cos (x) \end{array} |$

Step 4

Multiply $0$ by $| \begin{array}{c} 1 & \cos (x) \\ 1 & - \sin (x) \end{array} |$ .

$0 + 0 | \begin{array}{c} 1 & \sin (x) \\ 1 & - \sin (x) \end{array} | - \sec (x) | \begin{array}{c} 1 & \sin (x) \\ 1 & \cos (x) \end{array} |$

Step 5

Multiply $0$ by $| \begin{array}{c} 1 & \sin (x) \\ 1 & - \sin (x) \end{array} |$ .

$0 + 0 - \sec (x) | \begin{array}{c} 1 & \sin (x) \\ 1 & \cos (x) \end{array} |$

Step 6

Evaluate $| \begin{array}{c} 1 & \sin (x) \\ 1 & \cos (x) \end{array} |$ .

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Step 6.1

The determinant of a $2 \times 2$ matrix can be found using the formula $| \begin{array}{c} a & b \\ c & d \end{array} | = a d - c b$ .

$0 + 0 - \sec (x) (1 \cos (x) - \sin (x))$

Step 6.2

Multiply $\cos (x)$ by $1$ .

$0 + 0 - \sec (x) (\cos (x) - \sin (x))$

Step 7

Simplify the determinant.

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Step 7.1

Combine the opposite terms in $0 + 0 - \sec (x) (\cos (x) - \sin (x))$ .

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Step 7.1.1

Add $0$ and $0$ .

$0 - \sec (x) (\cos (x) - \sin (x))$

Step 7.1.2

Subtract $\sec (x) (\cos (x) - \sin (x))$ from $0$ .

$- \sec (x) (\cos (x) - \sin (x))$

Step 7.2

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

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

Step 7.3

Apply the distributive property.

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

Step 7.4

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

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Step 7.4.1

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

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

Step 7.4.2

Cancel the common factor.

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

Step 7.4.3

Rewrite the expression.

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

Step 7.5

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

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Step 7.5.1

Multiply $- 1$ by $- 1$ .

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

Step 7.5.2

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

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

Step 7.5.3

Combine $\frac{1}{\cos (x)}$ and $\sin (x)$ .

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

Step 7.6

Convert from $\frac{\sin (x)}{\cos (x)}$ to $\tan (x)$ .

$- 1 + \tan (x)$