Algebra Examples

${(\tan (x) + \sec (x))}^{2} = \frac{A + \sin (x)}{B - \sin (x)}$

Step 1

Rewrite the equation as $\frac{A + \sin (x)}{B - \sin (x)} = {(\tan (x) + \sec (x))}^{2}$ .

$\frac{A + \sin (x)}{B - \sin (x)} = {(\tan (x) + \sec (x))}^{2}$

Step 2

Find the LCD of the terms in the equation.

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Step 2.1

Finding the LCD of a list of values is the same as finding the LCM of the denominators of those values.

$B - \sin (x), 1$

Step 2.2

Remove parentheses.

$B - \sin (x), 1$

Step 2.3

The LCM of one and any expression is the expression.

$B - \sin (x)$

Step 3

Multiply each term in $\frac{A + \sin (x)}{B - \sin (x)} = {(\tan (x) + \sec (x))}^{2}$ by $B - \sin (x)$ to eliminate the fractions.

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Step 3.1

Multiply each term in $\frac{A + \sin (x)}{B - \sin (x)} = {(\tan (x) + \sec (x))}^{2}$ by $B - \sin (x)$ .

$\frac{A + \sin (x)}{B - \sin (x)} (B - \sin (x)) = {(\tan (x) + \sec (x))}^{2} (B - \sin (x))$

Step 3.2

Simplify the left side.

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Step 3.2.1

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

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Step 3.2.1.1

Cancel the common factor.

$\frac{A + \sin (x)}{B - \sin (x)} (B - \sin (x)) = {(\tan (x) + \sec (x))}^{2} (B - \sin (x))$

Step 3.2.1.2

Rewrite the expression.

$A + \sin (x) = {(\tan (x) + \sec (x))}^{2} (B - \sin (x))$

Step 3.3

Simplify the right side.

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Step 3.3.1

Simplify each term.

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Step 3.3.1.1

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

$A + \sin (x) = {(\frac{\sin (x)}{\cos (x)} + \sec (x))}^{2} (B - \sin (x))$

Step 3.3.1.2

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

$A + \sin (x) = {(\frac{\sin (x)}{\cos (x)} + \frac{1}{\cos (x)})}^{2} (B - \sin (x))$

Step 3.3.2

Simplify by multiplying through.

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Step 3.3.2.1

Apply the distributive property.

$A + \sin (x) = {(\frac{\sin (x)}{\cos (x)} + \frac{1}{\cos (x)})}^{2} B + {(\frac{\sin (x)}{\cos (x)} + \frac{1}{\cos (x)})}^{2} (- \sin (x))$

Step 3.3.2.2

Rewrite using the commutative property of multiplication.

$A + \sin (x) = {(\frac{\sin (x)}{\cos (x)} + \frac{1}{\cos (x)})}^{2} B - {(\frac{\sin (x)}{\cos (x)} + \frac{1}{\cos (x)})}^{2} \sin (x)$

Step 3.3.3

Simplify each term.

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Step 3.3.3.1

Simplify each term.

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Step 3.3.3.1.1

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

$A + \sin (x) = {(\tan (x) + \frac{1}{\cos (x)})}^{2} B - {(\frac{\sin (x)}{\cos (x)} + \frac{1}{\cos (x)})}^{2} \sin (x)$

Step 3.3.3.1.2

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

$A + \sin (x) = {(\tan (x) + \sec (x))}^{2} B - {(\frac{\sin (x)}{\cos (x)} + \frac{1}{\cos (x)})}^{2} \sin (x)$

Step 3.3.3.2

Simplify each term.

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Step 3.3.3.2.1

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

$A + \sin (x) = {(\tan (x) + \sec (x))}^{2} B - {(\tan (x) + \frac{1}{\cos (x)})}^{2} \sin (x)$

Step 3.3.3.2.2

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

$A + \sin (x) = {(\tan (x) + \sec (x))}^{2} B - {(\tan (x) + \sec (x))}^{2} \sin (x)$

Step 3.3.4

Reorder factors in ${(\tan (x) + \sec (x))}^{2} B - {(\tan (x) + \sec (x))}^{2} \sin (x)$ .

$A + \sin (x) = B {(\tan (x) + \sec (x))}^{2} - \sin (x) {(\tan (x) + \sec (x))}^{2}$

Step 4

Solve the equation.

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Step 4.1

Rewrite the equation as $B {(\tan (x) + \sec (x))}^{2} - \sin (x) {(\tan (x) + \sec (x))}^{2} = A + \sin (x)$ .

$B {(\tan (x) + \sec (x))}^{2} - \sin (x) {(\tan (x) + \sec (x))}^{2} = A + \sin (x)$

Step 4.2

Move all terms not containing $B$ to the right side of the equation.

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Step 4.2.1

Add $\sin (x) {(\tan (x) + \sec (x))}^{2}$ to both sides of the equation.

$B {(\tan (x) + \sec (x))}^{2} = A + \sin (x) + \sin (x) {(\tan (x) + \sec (x))}^{2}$

Step 4.2.2

Simplify each term.

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Step 4.2.2.1

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

$B {(\tan (x) + \sec (x))}^{2} = A + \sin (x) + \sin (x) {(\frac{\sin (x)}{\cos (x)} + \sec (x))}^{2}$

Step 4.2.2.2

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

$B {(\tan (x) + \sec (x))}^{2} = A + \sin (x) + \sin (x) {(\frac{\sin (x)}{\cos (x)} + \frac{1}{\cos (x)})}^{2}$

Step 4.2.3

Simplify each term.

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Step 4.2.3.1

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

$B {(\tan (x) + \sec (x))}^{2} = A + \sin (x) + \sin (x) {(\tan (x) + \frac{1}{\cos (x)})}^{2}$

Step 4.2.3.2

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

$B {(\tan (x) + \sec (x))}^{2} = A + \sin (x) + \sin (x) {(\tan (x) + \sec (x))}^{2}$

Step 4.3

Divide each term in $B {(\tan (x) + \sec (x))}^{2} = A + \sin (x) + \sin (x) {(\tan (x) + \sec (x))}^{2}$ by ${(\tan (x) + \sec (x))}^{2}$ and simplify.

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Step 4.3.1

Divide each term in $B {(\tan (x) + \sec (x))}^{2} = A + \sin (x) + \sin (x) {(\tan (x) + \sec (x))}^{2}$ by ${(\tan (x) + \sec (x))}^{2}$ .

$\frac{B {(\tan (x) + \sec (x))}^{2}}{{(\tan (x) + \sec (x))}^{2}} = \frac{A}{{(\tan (x) + \sec (x))}^{2}} + \frac{\sin (x)}{{(\tan (x) + \sec (x))}^{2}} + \frac{\sin (x) {(\tan (x) + \sec (x))}^{2}}{{(\tan (x) + \sec (x))}^{2}}$

Step 4.3.2

Simplify the left side.

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Step 4.3.2.1

Cancel the common factor of ${(\tan (x) + \sec (x))}^{2}$ .

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Step 4.3.2.1.1

Cancel the common factor.

Step 4.3.2.1.2

Divide $B$ by $1$ .

$B = \frac{A}{{(\tan (x) + \sec (x))}^{2}} + \frac{\sin (x)}{{(\tan (x) + \sec (x))}^{2}} + \frac{\sin (x) {(\tan (x) + \sec (x))}^{2}}{{(\tan (x) + \sec (x))}^{2}}$

Step 4.3.3

Simplify the right side.

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Step 4.3.3.1

Simplify each term.

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Step 4.3.3.1.1

Simplify the denominator.

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Step 4.3.3.1.1.1

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

$B = \frac{A}{{(\frac{\sin (x)}{\cos (x)} + \sec (x))}^{2}} + \frac{\sin (x)}{{(\tan (x) + \sec (x))}^{2}} + \frac{\sin (x) {(\tan (x) + \sec (x))}^{2}}{{(\tan (x) + \sec (x))}^{2}}$

Step 4.3.3.1.1.2

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

$B = \frac{A}{{(\frac{\sin (x)}{\cos (x)} + \frac{1}{\cos (x)})}^{2}} + \frac{\sin (x)}{{(\tan (x) + \sec (x))}^{2}} + \frac{\sin (x) {(\tan (x) + \sec (x))}^{2}}{{(\tan (x) + \sec (x))}^{2}}$

Step 4.3.3.1.2

Simplify the denominator.

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Step 4.3.3.1.2.1

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

$B = \frac{A}{{(\frac{\sin (x)}{\cos (x)} + \frac{1}{\cos (x)})}^{2}} + \frac{\sin (x)}{{(\frac{\sin (x)}{\cos (x)} + \sec (x))}^{2}} + \frac{\sin (x) {(\tan (x) + \sec (x))}^{2}}{{(\tan (x) + \sec (x))}^{2}}$

Step 4.3.3.1.2.2

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

$B = \frac{A}{{(\frac{\sin (x)}{\cos (x)} + \frac{1}{\cos (x)})}^{2}} + \frac{\sin (x)}{{(\frac{\sin (x)}{\cos (x)} + \frac{1}{\cos (x)})}^{2}} + \frac{\sin (x) {(\tan (x) + \sec (x))}^{2}}{{(\tan (x) + \sec (x))}^{2}}$

Step 4.3.3.1.3

Cancel the common factor of ${(\tan (x) + \sec (x))}^{2}$ .

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Step 4.3.3.1.3.1

Cancel the common factor.

Step 4.3.3.1.3.2

Divide $\sin (x)$ by $1$ .

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

Step 4.3.3.2

Simplify each term.

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Step 4.3.3.2.1

Simplify each term.

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Step 4.3.3.2.1.1

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

$B = \frac{A}{{(\tan (x) + \frac{1}{\cos (x)})}^{2}} + \frac{\sin (x)}{{(\frac{\sin (x)}{\cos (x)} + \frac{1}{\cos (x)})}^{2}} + \sin (x)$

Step 4.3.3.2.1.2

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

$B = \frac{A}{{(\tan (x) + \sec (x))}^{2}} + \frac{\sin (x)}{{(\frac{\sin (x)}{\cos (x)} + \frac{1}{\cos (x)})}^{2}} + \sin (x)$

Step 4.3.3.2.2

Simplify each term.

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Step 4.3.3.2.2.1

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

$B = \frac{A}{{(\tan (x) + \sec (x))}^{2}} + \frac{\sin (x)}{{(\tan (x) + \frac{1}{\cos (x)})}^{2}} + \sin (x)$

Step 4.3.3.2.2.2

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

$B = \frac{A}{{(\tan (x) + \sec (x))}^{2}} + \frac{\sin (x)}{{(\tan (x) + \sec (x))}^{2}} + \sin (x)$