Trigonometry Examples

$\csc (x) (\sin (5 x) - \sin (3 x))$

Step 1

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

$\frac{1}{\sin (x)} (\sin (5 x) - \sin (3 x))$

Step 2

Simplify terms.

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Step 2.1

Apply the distributive property.

$\frac{1}{\sin (x)} \sin (5 x) + \frac{1}{\sin (x)} (- \sin (3 x))$

Step 2.2

Combine $\frac{1}{\sin (x)}$ and $\sin (5 x)$ .

$\frac{\sin (5 x)}{\sin (x)} + \frac{1}{\sin (x)} (- \sin (3 x))$

Step 2.3

Rewrite using the commutative property of multiplication.

$\frac{\sin (5 x)}{\sin (x)} - \frac{1}{\sin (x)} \sin (3 x)$

Step 2.4

Combine $\sin (3 x)$ and $\frac{1}{\sin (x)}$ .

$\frac{\sin (5 x)}{\sin (x)} - \frac{\sin (3 x)}{\sin (x)}$

Step 3

Simplify each term.

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Step 3.1

Rewrite $\frac{\sin (5 x)}{\sin (x)}$ as a product.

$\sin (5 x) \frac{1}{\sin (x)} - \frac{\sin (3 x)}{\sin (x)}$

Step 3.2

Write $\sin (5 x)$ as a fraction with denominator $1$ .

$\frac{\sin (5 x)}{1} \cdot \frac{1}{\sin (x)} - \frac{\sin (3 x)}{\sin (x)}$

Step 3.3

Simplify.

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Step 3.3.1

Divide $\sin (5 x)$ by $1$ .

$\sin (5 x) \frac{1}{\sin (x)} - \frac{\sin (3 x)}{\sin (x)}$

Step 3.3.2

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

$\sin (5 x) \csc (x) - \frac{\sin (3 x)}{\sin (x)}$

Step 3.4

Rewrite $\frac{\sin (3 x)}{\sin (x)}$ as a product.

$\sin (5 x) \csc (x) - (\sin (3 x) \frac{1}{\sin (x)})$

Step 3.5

Write $\sin (3 x)$ as a fraction with denominator $1$ .

$\sin (5 x) \csc (x) - (\frac{\sin (3 x)}{1} \cdot \frac{1}{\sin (x)})$

Step 3.6

Simplify.

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Step 3.6.1

Divide $\sin (3 x)$ by $1$ .

$\sin (5 x) \csc (x) - (\sin (3 x) \frac{1}{\sin (x)})$

Step 3.6.2

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

$\sin (5 x) \csc (x) - (\sin (3 x) \csc (x))$

$\sin (5 x) \csc (x) - \sin (3 x) \csc (x)$