Trigonometry Examples

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

Step 1

Multiply both sides of the equation by $\cos (x)$ .

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

Step 2

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

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Step 2.1

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

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

Step 2.2

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

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

Step 2.3

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

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

Step 2.4

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

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

Step 2.5

Add $1$ and $1$ .

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

Step 3

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

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Step 3.1

Cancel the common factor.

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

Step 3.2

Rewrite the expression.

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

Step 4

Move all the expressions to the left side of the equation.

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Step 4.1

Subtract $1$ from both sides of the equation.

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

Step 4.2

Add $\sin (x)$ to both sides of the equation.

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

Step 5

Simplify $\frac{\cos^{2} (x)}{1 + \sin (x)} - 1 + \sin (x)$ .

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Step 5.1

To write $\sin (x)$ as a fraction with a common denominator, multiply by $\frac{1 + \sin (x)}{1 + \sin (x)}$ .

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

Step 5.2

Combine the numerators over the common denominator.

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

Step 5.3

Simplify each term.

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Step 5.3.1

Simplify the numerator.

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Step 5.3.1.1

Apply the distributive property.

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

Step 5.3.1.2

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

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

Step 5.3.1.3

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

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Step 5.3.1.3.1

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

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

Step 5.3.1.3.2

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

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

Step 5.3.1.3.3

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

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

Step 5.3.1.3.4

Add $1$ and $1$ .

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

Step 5.3.1.4

Rewrite $\cos^{2} (x) + \sin (x) + \sin^{2} (x)$ in a factored form.

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Step 5.3.1.4.1

Rearrange terms.

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

Step 5.3.1.4.2

Rearrange terms.

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

Step 5.3.1.4.3

Apply pythagorean identity.

$\frac{\sin (x) + 1}{1 + \sin (x)} - 1 = 0$

Step 5.3.2

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

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Step 5.3.2.1

Reorder terms.

$\frac{1 + \sin (x)}{1 + \sin (x)} - 1 = 0$

Step 5.3.2.2

Cancel the common factor.

$\frac{1 + \sin (x)}{1 + \sin (x)} - 1 = 0$

Step 5.3.2.3

Rewrite the expression.

$1 - 1 = 0$

Step 5.4

Subtract $1$ from $1$ .

$0 = 0$

Step 6

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

All real numbers

Step 7

The result can be shown in multiple forms.

All real numbers

Interval Notation:

$(- \infty, \infty)$