Trigonometry Examples

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

Step 1

Simplify the left side.

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Step 1.1

Simplify $\frac{1}{\csc (x) + 1} + \frac{1}{\sin (x) + 1}$ .

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Step 1.1.1

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

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

Step 1.1.2

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

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

Step 1.1.3

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

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

Step 1.1.4

Write each expression with a common denominator of $(\frac{1}{\sin (x)} + 1) (\sin (x) + 1)$ , by multiplying each by an appropriate factor of $1$ .

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Step 1.1.4.1

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

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

Step 1.1.4.2

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

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

Step 1.1.4.3

Reorder the factors of $(\frac{1}{\sin (x)} + 1) (\sin (x) + 1)$ .

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

Step 1.1.5

Combine the numerators over the common denominator.

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

Step 1.1.6

Add $1$ and $1$ .

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

Step 1.1.7

Convert from $\frac{1}{\sin (x)}$ to $\csc (x)$ .

$\frac{\sin (x) + 2 + \csc (x)}{(\sin (x) + 1) (\frac{1}{\sin (x)} + 1)} = 1$

Step 1.1.8

Convert from $\frac{1}{\sin (x)}$ to $\csc (x)$ .

$\frac{\sin (x) + 2 + \csc (x)}{(\sin (x) + 1) (\csc (x) + 1)} = 1$

Step 2

Multiply both sides by $(\sin (x) + 1) (\csc (x) + 1)$ .

$\frac{\sin (x) + 2 + \csc (x)}{(\sin (x) + 1) (\csc (x) + 1)} ((\sin (x) + 1) (\csc (x) + 1)) = 1 ((\sin (x) + 1) (\csc (x) + 1))$

Step 3

Simplify.

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Step 3.1

Simplify the left side.

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Step 3.1.1

Simplify $\frac{\sin (x) + 2 + \csc (x)}{(\sin (x) + 1) (\csc (x) + 1)} ((\sin (x) + 1) (\csc (x) + 1))$ .

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Step 3.1.1.1

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

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Step 3.1.1.1.1

Cancel the common factor.

$\frac{\sin (x) + 2 + \csc (x)}{(\sin (x) + 1) (\csc (x) + 1)} ((\sin (x) + 1) (\csc (x) + 1)) = 1 ((\sin (x) + 1) (\csc (x) + 1))$

Step 3.1.1.1.2

Rewrite the expression.

$\sin (x) + 2 + \csc (x) = 1 ((\sin (x) + 1) (\csc (x) + 1))$

Step 3.1.1.2

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

$\sin (x) + 2 + \frac{1}{\sin (x)} = 1 ((\sin (x) + 1) (\csc (x) + 1))$

Step 3.1.1.3

Convert from $\frac{1}{\sin (x)}$ to $\csc (x)$ .

$\sin (x) + 2 + \csc (x) = 1 ((\sin (x) + 1) (\csc (x) + 1))$

Step 3.2

Simplify the right side.

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Step 3.2.1

Simplify $1 ((\sin (x) + 1) (\csc (x) + 1))$ .

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Step 3.2.1.1

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

$\sin (x) + 2 + \csc (x) = (\sin (x) + 1) (\csc (x) + 1)$

Step 3.2.1.2

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

$\sin (x) + 2 + \csc (x) = (\sin (x) + 1) (\frac{1}{\sin (x)} + 1)$

Step 3.2.1.3

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

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Step 3.2.1.3.1

Apply the distributive property.

$\sin (x) + 2 + \csc (x) = \sin (x) (\frac{1}{\sin (x)} + 1) + 1 (\frac{1}{\sin (x)} + 1)$

Step 3.2.1.3.2

Apply the distributive property.

$\sin (x) + 2 + \csc (x) = \sin (x) \frac{1}{\sin (x)} + \sin (x) \cdot 1 + 1 (\frac{1}{\sin (x)} + 1)$

Step 3.2.1.3.3

Apply the distributive property.

$\sin (x) + 2 + \csc (x) = \sin (x) \frac{1}{\sin (x)} + \sin (x) \cdot 1 + 1 \frac{1}{\sin (x)} + 1 \cdot 1$

Step 3.2.1.4

Simplify and combine like terms.

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Step 3.2.1.4.1

Simplify each term.

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Step 3.2.1.4.1.1

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

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Step 3.2.1.4.1.1.1

Cancel the common factor.

$\sin (x) + 2 + \csc (x) = \sin (x) \frac{1}{\sin (x)} + \sin (x) \cdot 1 + 1 \frac{1}{\sin (x)} + 1 \cdot 1$

Step 3.2.1.4.1.1.2

Rewrite the expression.

$\sin (x) + 2 + \csc (x) = 1 + \sin (x) \cdot 1 + 1 \frac{1}{\sin (x)} + 1 \cdot 1$

Step 3.2.1.4.1.2

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

$\sin (x) + 2 + \csc (x) = 1 + \sin (x) + 1 \frac{1}{\sin (x)} + 1 \cdot 1$

Step 3.2.1.4.1.3

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

$\sin (x) + 2 + \csc (x) = 1 + \sin (x) + \frac{1}{\sin (x)} + 1 \cdot 1$

Step 3.2.1.4.1.4

Multiply $1$ by $1$ .

$\sin (x) + 2 + \csc (x) = 1 + \sin (x) + \frac{1}{\sin (x)} + 1$

Step 3.2.1.4.2

Add $1$ and $1$ .

$\sin (x) + 2 + \csc (x) = 2 + \sin (x) + \frac{1}{\sin (x)}$

Step 3.2.1.5

Convert from $\frac{1}{\sin (x)}$ to $\csc (x)$ .

$\sin (x) + 2 + \csc (x) = 2 + \sin (x) + \csc (x)$

Step 4

Solve for $x$ .

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Step 4.1

Move all terms containing $x$ to the left side of the equation.

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Step 4.1.1

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

$\sin (x) + 2 + \csc (x) - \sin (x) = 2 + \csc (x)$

Step 4.1.2

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

$\sin (x) + 2 + \csc (x) - \sin (x) - \csc (x) = 2$

Step 4.1.3

Combine the opposite terms in $\sin (x) + 2 + \csc (x) - \sin (x) - \csc (x)$ .

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Step 4.1.3.1

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

$0 + 2 + \csc (x) - \csc (x) = 2$

Step 4.1.3.2

Add $0$ and $2$ .

$2 + \csc (x) - \csc (x) = 2$

Step 4.1.3.3

Subtract $\csc (x)$ from $\csc (x)$ .

$2 + 0 = 2$

Step 4.1.3.4

Add $2$ and $0$ .

$2 = 2$

Step 4.2

Since $2 = 2$ , 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:

$(- \infty, \infty)$