Trigonometry Examples

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

Step 1

Start on the left side.

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

Step 2

Apply the reciprocal identity to $\csc (x)$ .

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

Step 3

Simplify.

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Step 3.1

Simplify each term.

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Step 3.1.1

Simplify the denominator.

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Step 3.1.1.1

Write $1$ as a fraction with a common denominator.

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

Step 3.1.1.2

Combine the numerators over the common denominator.

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

Step 3.1.2

Multiply the numerator by the reciprocal of the denominator.

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

Step 3.1.3

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

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

Step 3.2

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

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

Step 3.3

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

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

Step 3.4

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

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Step 3.4.1

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

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

Step 3.4.2

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

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

Step 3.4.3

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

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

Step 3.5

Combine the numerators over the common denominator.

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

Step 3.6

Simplify the numerator.

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Step 3.6.1

Apply the distributive property.

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

Step 3.6.2

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

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Step 3.6.2.1

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

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

Step 3.6.2.2

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

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

Step 3.6.2.3

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

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

Step 3.6.2.4

Add $1$ and $1$ .

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

Step 3.6.3

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

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

Step 3.6.4

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

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

Step 3.6.5

Factor using the perfect square rule.

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Step 3.6.5.1

Rearrange terms.

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

Step 3.6.5.2

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

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

Step 3.6.5.3

Check that the middle term is two times the product of the numbers being squared in the first term and third term.

$2 \sin (x) = 2 \cdot 1 \cdot \sin (x)$

Step 3.6.5.4

Rewrite the polynomial.

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

Step 3.6.5.5

Factor using the perfect square trinomial rule $a^{2} + 2 a b + b^{2} = {(a + b)}^{2}$ , where $a = 1$ and $b = \sin (x)$ .

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

Step 3.7

Cancel the common factor of ${(1 + \sin (x))}^{2}$ and $1 + \sin (x)$ .

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

Step 4

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

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Step 4.1

Cancel the common factor.

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

Step 4.2

Rewrite the expression.

$1$

Step 5

Because the two sides have been shown to be equivalent, the equation is an identity.

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