Trigonometry Examples

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

Step 1

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

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

Step 2

Multiply the numerator and denominator of the fraction by $\sin (y)$ .

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Step 2.1

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

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

Step 2.2

Combine.

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

Step 3

Apply the distributive property.

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

Step 4

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

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Step 4.1

Cancel the common factor.

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

Step 4.2

Rewrite the expression.

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

Step 5

Simplify the numerator.

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Step 5.1

Multiply $\sin (y) \sin (y)$ .

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Step 5.1.1

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

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

Step 5.1.2

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

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

Step 5.1.3

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

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

Step 5.1.4

Add $1$ and $1$ .

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

Step 5.2

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

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

Step 5.3

Factor $\sin (y)$ out of $\sin^{2} (y) + \sin (y)$ .

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Step 5.3.1

Factor $\sin (y)$ out of $\sin^{2} (y)$ .

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

Step 5.3.2

Multiply by $1$ .

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

Step 5.3.3

Factor $\sin (y)$ out of $\sin (y) \sin (y) + \sin (y) \cdot 1$ .

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

Step 6

Reduce the expression by cancelling the common factors.

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Step 6.1

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

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

Step 6.2

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

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Step 6.2.1

Reorder terms.

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

Step 6.2.2

Cancel the common factor.

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

Step 6.2.3

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

$\sin (y)$