Trigonometry Examples

$\frac{4 \sin (x)}{1 - \sin^{2} (x)} = \frac{1 + \sin (x)}{1 - \sin (x)} + \frac{\sin (x) - 1}{1 + \sin (x)}$

Step 1

Factor each term.

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Step 1.1

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

$\frac{4 \sin (x)}{1^{2} - \sin^{2} (x)} = \frac{1 + \sin (x)}{1 - \sin (x)} + \frac{\sin (x) - 1}{1 + \sin (x)}$

Step 1.2

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = 1$ and $b = \sin (x)$ .

$\frac{4 \sin (x)}{(1 + \sin (x)) (1 - \sin (x))} = \frac{1 + \sin (x)}{1 - \sin (x)} + \frac{\sin (x) - 1}{1 + \sin (x)}$

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.

$(1 + \sin (x)) (1 - \sin (x)), 1 - \sin (x), 1 + \sin (x)$

Step 2.2

The LCM is the smallest positive number that all of the numbers divide into evenly.

1. List the prime factors of each number.

2. Multiply each factor the greatest number of times it occurs in either number.

Step 2.3

The number $1$ is not a prime number because it only has one positive factor, which is itself.

Not prime

Step 2.4

The LCM of $1, 1, 1$ is the result of multiplying all prime factors the greatest number of times they occur in either number.

$1$

Step 2.5

The factor for $1 + \sin (x)$ is $1 + \sin (x)$ itself.

$(1 + \sin (x)) = 1 + \sin (x)$

$(1 + \sin (x))$ occurs $1$ time.

Step 2.6

The factor for $1 - \sin (x)$ is $1 - \sin (x)$ itself.

$(1 - \sin (x)) = 1 - \sin (x)$

$(1 - \sin (x))$ occurs $1$ time.

Step 2.7

The factor for $1 + \sin (x)$ is $1 + \sin (x)$ itself.

$(1 + \sin (x)) = 1 + \sin (x)$

$(1 + \sin (x))$ occurs $1$ time.

Step 2.8

The LCM of $1 + \sin (x), 1 - \sin (x), 1 - \sin (x), 1 + \sin (x)$ is the result of multiplying all factors the greatest number of times they occur in either term.

$(1 + \sin (x)) (1 - \sin (x))$

Step 3

Multiply each term in $\frac{4 \sin (x)}{(1 + \sin (x)) (1 - \sin (x))} = \frac{1 + \sin (x)}{1 - \sin (x)} + \frac{\sin (x) - 1}{1 + \sin (x)}$ by $(1 + \sin (x)) (1 - \sin (x))$ to eliminate the fractions.

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Step 3.1

Multiply each term in $\frac{4 \sin (x)}{(1 + \sin (x)) (1 - \sin (x))} = \frac{1 + \sin (x)}{1 - \sin (x)} + \frac{\sin (x) - 1}{1 + \sin (x)}$ by $(1 + \sin (x)) (1 - \sin (x))$ .

$\frac{4 \sin (x)}{(1 + \sin (x)) (1 - \sin (x))} ((1 + \sin (x)) (1 - \sin (x))) = \frac{1 + \sin (x)}{1 - \sin (x)} ((1 + \sin (x)) (1 - \sin (x))) + \frac{\sin (x) - 1}{1 + \sin (x)} ((1 + \sin (x)) (1 - \sin (x)))$

Step 3.2

Simplify the left side.

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Step 3.2.1

Cancel the common factor of $(1 + \sin (x)) (1 - \sin (x))$ .

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Step 3.2.1.1

Cancel the common factor.

Step 3.2.1.2

Rewrite the expression.

$4 \sin (x) = \frac{1 + \sin (x)}{1 - \sin (x)} ((1 + \sin (x)) (1 - \sin (x))) + \frac{\sin (x) - 1}{1 + \sin (x)} ((1 + \sin (x)) (1 - \sin (x)))$

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

Cancel the common factor of $1 - \sin (x)$ .

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Step 3.3.1.1.1

Factor $1 - \sin (x)$ out of $(1 + \sin (x)) (1 - \sin (x))$ .

$4 \sin (x) = \frac{1 + \sin (x)}{1 - \sin (x)} ((1 - \sin (x)) (1 + \sin (x))) + \frac{\sin (x) - 1}{1 + \sin (x)} ((1 + \sin (x)) (1 - \sin (x)))$

Step 3.3.1.1.2

Cancel the common factor.

$4 \sin (x) = \frac{1 + \sin (x)}{1 - \sin (x)} ((1 - \sin (x)) (1 + \sin (x))) + \frac{\sin (x) - 1}{1 + \sin (x)} ((1 + \sin (x)) (1 - \sin (x)))$

Step 3.3.1.1.3

Rewrite the expression.

$4 \sin (x) = (1 + \sin (x)) (1 + \sin (x)) + \frac{\sin (x) - 1}{1 + \sin (x)} ((1 + \sin (x)) (1 - \sin (x)))$

Step 3.3.1.2

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

$4 \sin (x) = {(1 + \sin (x))}^{1} (1 + \sin (x)) + \frac{\sin (x) - 1}{1 + \sin (x)} ((1 + \sin (x)) (1 - \sin (x)))$

Step 3.3.1.3

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

$4 \sin (x) = {(1 + \sin (x))}^{1} {(1 + \sin (x))}^{1} + \frac{\sin (x) - 1}{1 + \sin (x)} ((1 + \sin (x)) (1 - \sin (x)))$

Step 3.3.1.4

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

$4 \sin (x) = {(1 + \sin (x))}^{1 + 1} + \frac{\sin (x) - 1}{1 + \sin (x)} ((1 + \sin (x)) (1 - \sin (x)))$

Step 3.3.1.5

Add $1$ and $1$ .

$4 \sin (x) = {(1 + \sin (x))}^{2} + \frac{\sin (x) - 1}{1 + \sin (x)} ((1 + \sin (x)) (1 - \sin (x)))$

Step 3.3.1.6

Cancel the common factor of $1 + \sin (x)$ .

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Step 3.3.1.6.1

Cancel the common factor.

$4 \sin (x) = {(1 + \sin (x))}^{2} + \frac{\sin (x) - 1}{1 + \sin (x)} ((1 + \sin (x)) (1 - \sin (x)))$

Step 3.3.1.6.2

Rewrite the expression.

$4 \sin (x) = {(1 + \sin (x))}^{2} + (\sin (x) - 1) (1 - \sin (x))$

Step 3.3.1.7

Factor $- 1$ out of $\sin (x)$ .

$4 \sin (x) = {(1 + \sin (x))}^{2} + (- 1 (- \sin (x)) - 1) (1 - \sin (x))$

Step 3.3.1.8

Rewrite $- 1$ as $- 1 (1)$ .

$4 \sin (x) = {(1 + \sin (x))}^{2} + (- 1 (- \sin (x)) - 1 (1)) (1 - \sin (x))$

Step 3.3.1.9

Factor $- 1$ out of $- 1 (- \sin (x)) - 1 (1)$ .

$4 \sin (x) = {(1 + \sin (x))}^{2} - 1 (- \sin (x) + 1) (1 - \sin (x))$

Step 3.3.1.10

Reorder terms.

$4 \sin (x) = {(1 + \sin (x))}^{2} - 1 (- \sin (x) + 1) (- \sin (x) + 1)$

Step 3.3.1.11

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

$4 \sin (x) = {(1 + \sin (x))}^{2} - 1 ({(- \sin (x) + 1)}^{1} (- \sin (x) + 1))$

Step 3.3.1.12

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

$4 \sin (x) = {(1 + \sin (x))}^{2} - 1 ({(- \sin (x) + 1)}^{1} {(- \sin (x) + 1)}^{1})$

Step 3.3.1.13

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

$4 \sin (x) = {(1 + \sin (x))}^{2} - 1 {(- \sin (x) + 1)}^{1 + 1}$

Step 3.3.1.14

Add $1$ and $1$ .

$4 \sin (x) = {(1 + \sin (x))}^{2} - 1 {(- \sin (x) + 1)}^{2}$

Step 3.3.1.15

Rewrite $- 1 {(- \sin (x) + 1)}^{2}$ as $- {(- \sin (x) + 1)}^{2}$ .

$4 \sin (x) = {(1 + \sin (x))}^{2} - {(- \sin (x) + 1)}^{2}$

Step 4

Solve the equation.

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Step 4.1

Since $\sin (x)$ is on the right side of the equation, switch the sides so it is on the left side of the equation.

${(1 + \sin (x))}^{2} - {(- \sin (x) + 1)}^{2} = 4 \sin (x)$

Step 4.2

Move all terms containing $\sin (x)$ to the left side of the equation.

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Step 4.2.1

Subtract $4 \sin (x)$ from both sides of the equation.

${(1 + \sin (x))}^{2} - {(- \sin (x) + 1)}^{2} - 4 \sin (x) = 0$

Step 4.2.2

Simplify each term.

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Step 4.2.2.1

Rewrite ${(1 + \sin (x))}^{2}$ as $(1 + \sin (x)) (1 + \sin (x))$ .

$(1 + \sin (x)) (1 + \sin (x)) - {(- \sin (x) + 1)}^{2} - 4 \sin (x) = 0$

Step 4.2.2.2

Expand $(1 + \sin (x)) (1 + \sin (x))$ using the FOIL Method.

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Step 4.2.2.2.1

Apply the distributive property.

$1 (1 + \sin (x)) + \sin (x) (1 + \sin (x)) - {(- \sin (x) + 1)}^{2} - 4 \sin (x) = 0$

Step 4.2.2.2.2

Apply the distributive property.

$1 \cdot 1 + 1 \sin (x) + \sin (x) (1 + \sin (x)) - {(- \sin (x) + 1)}^{2} - 4 \sin (x) = 0$

Step 4.2.2.2.3

Apply the distributive property.

$1 \cdot 1 + 1 \sin (x) + \sin (x) \cdot 1 + \sin (x) \sin (x) - {(- \sin (x) + 1)}^{2} - 4 \sin (x) = 0$

Step 4.2.2.3

Simplify and combine like terms.

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Step 4.2.2.3.1

Simplify each term.

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Step 4.2.2.3.1.1

Multiply $1$ by $1$ .

$1 + 1 \sin (x) + \sin (x) \cdot 1 + \sin (x) \sin (x) - {(- \sin (x) + 1)}^{2} - 4 \sin (x) = 0$

Step 4.2.2.3.1.2

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

$1 + \sin (x) + \sin (x) \cdot 1 + \sin (x) \sin (x) - {(- \sin (x) + 1)}^{2} - 4 \sin (x) = 0$

Step 4.2.2.3.1.3

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

$1 + \sin (x) + \sin (x) + \sin (x) \sin (x) - {(- \sin (x) + 1)}^{2} - 4 \sin (x) = 0$

Step 4.2.2.3.1.4

Multiply $\sin (x)$ by $\sin (x)$ .

$1 + \sin (x) + \sin (x) + \sin^{2} (x) - {(- \sin (x) + 1)}^{2} - 4 \sin (x) = 0$

Step 4.2.2.3.2

Add $\sin (x)$ and $\sin (x)$ .

$1 + 2 \sin (x) + \sin^{2} (x) - {(- \sin (x) + 1)}^{2} - 4 \sin (x) = 0$

Step 4.2.2.4

Rewrite ${(- \sin (x) + 1)}^{2}$ as $(- \sin (x) + 1) (- \sin (x) + 1)$ .

$1 + 2 \sin (x) + \sin^{2} (x) - ((- \sin (x) + 1) (- \sin (x) + 1)) - 4 \sin (x) = 0$

Step 4.2.2.5

Expand $(- \sin (x) + 1) (- \sin (x) + 1)$ using the FOIL Method.

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Step 4.2.2.5.1

Apply the distributive property.

$1 + 2 \sin (x) + \sin^{2} (x) - (- \sin (x) (- \sin (x) + 1) + 1 (- \sin (x) + 1)) - 4 \sin (x) = 0$

Step 4.2.2.5.2

Apply the distributive property.

$1 + 2 \sin (x) + \sin^{2} (x) - (- \sin (x) (- \sin (x)) - \sin (x) \cdot 1 + 1 (- \sin (x) + 1)) - 4 \sin (x) = 0$

Step 4.2.2.5.3

Apply the distributive property.

$1 + 2 \sin (x) + \sin^{2} (x) - (- \sin (x) (- \sin (x)) - \sin (x) \cdot 1 + 1 (- \sin (x)) + 1 \cdot 1) - 4 \sin (x) = 0$

Step 4.2.2.6

Simplify and combine like terms.

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Step 4.2.2.6.1

Simplify each term.

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Step 4.2.2.6.1.1

Rewrite using the commutative property of multiplication.

$1 + 2 \sin (x) + \sin^{2} (x) - (- 1 \cdot - 1 \sin (x) \sin (x) - \sin (x) \cdot 1 + 1 (- \sin (x)) + 1 \cdot 1) - 4 \sin (x) = 0$

Step 4.2.2.6.1.2

Multiply $\sin (x)$ by $\sin (x)$ by adding the exponents.

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Step 4.2.2.6.1.2.1

Move $\sin (x)$ .

$1 + 2 \sin (x) + \sin^{2} (x) - (- 1 \cdot - 1 (\sin (x) \sin (x)) - \sin (x) \cdot 1 + 1 (- \sin (x)) + 1 \cdot 1) - 4 \sin (x) = 0$

Step 4.2.2.6.1.2.2

Multiply $\sin (x)$ by $\sin (x)$ .

$1 + 2 \sin (x) + \sin^{2} (x) - (- 1 \cdot - 1 \sin^{2} (x) - \sin (x) \cdot 1 + 1 (- \sin (x)) + 1 \cdot 1) - 4 \sin (x) = 0$

Step 4.2.2.6.1.3

Multiply $- 1$ by $- 1$ .

$1 + 2 \sin (x) + \sin^{2} (x) - (1 \sin^{2} (x) - \sin (x) \cdot 1 + 1 (- \sin (x)) + 1 \cdot 1) - 4 \sin (x) = 0$

Step 4.2.2.6.1.4

Multiply $\sin^{2} (x)$ by $1$ .

$1 + 2 \sin (x) + \sin^{2} (x) - (\sin^{2} (x) - \sin (x) \cdot 1 + 1 (- \sin (x)) + 1 \cdot 1) - 4 \sin (x) = 0$

Step 4.2.2.6.1.5

Multiply $- 1$ by $1$ .

$1 + 2 \sin (x) + \sin^{2} (x) - (\sin^{2} (x) - \sin (x) + 1 (- \sin (x)) + 1 \cdot 1) - 4 \sin (x) = 0$

Step 4.2.2.6.1.6

Multiply $- \sin (x)$ by $1$ .

$1 + 2 \sin (x) + \sin^{2} (x) - (\sin^{2} (x) - \sin (x) - \sin (x) + 1 \cdot 1) - 4 \sin (x) = 0$

Step 4.2.2.6.1.7

Multiply $1$ by $1$ .

$1 + 2 \sin (x) + \sin^{2} (x) - (\sin^{2} (x) - \sin (x) - \sin (x) + 1) - 4 \sin (x) = 0$

Step 4.2.2.6.2

Subtract $\sin (x)$ from $- \sin (x)$ .

$1 + 2 \sin (x) + \sin^{2} (x) - (\sin^{2} (x) - 2 \sin (x) + 1) - 4 \sin (x) = 0$

Step 4.2.2.7

Apply the distributive property.

$1 + 2 \sin (x) + \sin^{2} (x) - \sin^{2} (x) - (- 2 \sin (x)) - 1 \cdot 1 - 4 \sin (x) = 0$

Step 4.2.2.8

Simplify.

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Step 4.2.2.8.1

Multiply $- 2$ by $- 1$ .

$1 + 2 \sin (x) + \sin^{2} (x) - \sin^{2} (x) + 2 \sin (x) - 1 \cdot 1 - 4 \sin (x) = 0$

Step 4.2.2.8.2

Multiply $- 1$ by $1$ .

$1 + 2 \sin (x) + \sin^{2} (x) - \sin^{2} (x) + 2 \sin (x) - 1 - 4 \sin (x) = 0$

Step 4.2.3

Combine the opposite terms in $1 + 2 \sin (x) + \sin^{2} (x) - \sin^{2} (x) + 2 \sin (x) - 1 - 4 \sin (x)$ .

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Step 4.2.3.1

Subtract $\sin^{2} (x)$ from $\sin^{2} (x)$ .

$1 + 2 \sin (x) + 0 + 2 \sin (x) - 1 - 4 \sin (x) = 0$

Step 4.2.3.2

Add $1 + 2 \sin (x)$ and $0$ .

$1 + 2 \sin (x) + 2 \sin (x) - 1 - 4 \sin (x) = 0$

Step 4.2.3.3

Subtract $1$ from $1$ .

$2 \sin (x) + 2 \sin (x) + 0 - 4 \sin (x) = 0$

Step 4.2.3.4

Add $2 \sin (x) + 2 \sin (x)$ and $0$ .

$2 \sin (x) + 2 \sin (x) - 4 \sin (x) = 0$

Step 4.2.4

Add $2 \sin (x)$ and $2 \sin (x)$ .

$4 \sin (x) - 4 \sin (x) = 0$

Step 4.2.5

Subtract $4 \sin (x)$ from $4 \sin (x)$ .

$0 = 0$

Step 4.3

Since $0 = 0$ , the equation will always be true for any value of $x$ .

All real numbers

Step 5

The result can be shown in multiple forms.

All real numbers

Interval Notation:

$(- \infty, \infty)$