Trigonometry Examples

$\frac{\sin (3 x) \cot (5 x)}{x \cot (4 x)}$

Step 1

Separate fractions.

$\frac{\cot (5 x)}{x} \cdot \frac{\sin (3 x)}{\cot (4 x)}$

Step 2

Rewrite $\cot (4 x)$ in terms of sines and cosines.

$\frac{\cot (5 x)}{x} \cdot \frac{\sin (3 x)}{\frac{\cos (4 x)}{\sin (4 x)}}$

Step 3

Multiply by the reciprocal of the fraction to divide by $\frac{\cos (4 x)}{\sin (4 x)}$ .

$\frac{\cot (5 x)}{x} (\sin (3 x) \frac{\sin (4 x)}{\cos (4 x)})$

Step 4

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

$\frac{\cot (5 x)}{x} (\frac{\sin (3 x)}{1} \cdot \frac{\sin (4 x)}{\cos (4 x)})$

Step 5

Simplify.

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Step 5.1

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

$\frac{\cot (5 x)}{x} (\sin (3 x) \frac{\sin (4 x)}{\cos (4 x)})$

Step 5.2

Combine $\sin (3 x)$ and $\frac{\sin (4 x)}{\cos (4 x)}$ .

$\frac{\cot (5 x)}{x} \cdot \frac{\sin (3 x) \sin (4 x)}{\cos (4 x)}$

Step 6

Rewrite $\cot (5 x)$ in terms of sines and cosines.

$\frac{\frac{\cos (5 x)}{\sin (5 x)}}{x} \cdot \frac{\sin (3 x) \sin (4 x)}{\cos (4 x)}$

Step 7

Multiply the numerator by the reciprocal of the denominator.

$\frac{\cos (5 x)}{\sin (5 x)} \cdot \frac{1}{x} \frac{\sin (3 x) \sin (4 x)}{\cos (4 x)}$

Step 8

Multiply $\frac{\cos (5 x)}{\sin (5 x)}$ by $\frac{1}{x}$ .

$\frac{\cos (5 x)}{\sin (5 x) x} \cdot \frac{\sin (3 x) \sin (4 x)}{\cos (4 x)}$

Step 9

Multiply $\frac{\cos (5 x)}{\sin (5 x) x}$ by $\frac{\sin (3 x) \sin (4 x)}{\cos (4 x)}$ .

$\frac{\cos (5 x) (\sin (3 x) \sin (4 x))}{\sin (5 x) x \cos (4 x)}$

Step 10

Separate fractions.

$\frac{\sin (3 x) \sin (4 x)}{x \cos (4 x)} \cdot \frac{\cos (5 x)}{\sin (5 x)}$

Step 11

Convert from $\frac{\cos (5 x)}{\sin (5 x)}$ to $\cot (5 x)$ .

$\frac{\sin (3 x) \sin (4 x)}{x \cos (4 x)} \cot (5 x)$

Step 12

Separate fractions.

$\frac{\sin (4 x)}{x} \cdot \frac{\sin (3 x)}{\cos (4 x)} \cot (5 x)$

Step 13

Rewrite $\frac{\sin (3 x)}{\cos (4 x)}$ as a product.

$\frac{\sin (4 x)}{x} (\sin (3 x) \frac{1}{\cos (4 x)}) \cot (5 x)$

Step 14

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

$\frac{\sin (4 x)}{x} (\frac{\sin (3 x)}{1} \cdot \frac{1}{\cos (4 x)}) \cot (5 x)$

Step 15

Simplify.

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Step 15.1

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

$\frac{\sin (4 x)}{x} (\sin (3 x) \frac{1}{\cos (4 x)}) \cot (5 x)$

Step 15.2

Convert from $\frac{1}{\cos (4 x)}$ to $\sec (4 x)$ .

$\frac{\sin (4 x)}{x} (\sin (3 x) \sec (4 x)) \cot (5 x)$

Step 16

Multiply $\frac{\sin (4 x)}{x} (\sin (3 x) \sec (4 x))$ .

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Step 16.1

Combine $\sin (3 x)$ and $\frac{\sin (4 x)}{x}$ .

$\frac{\sin (3 x) \sin (4 x)}{x} \sec (4 x) \cot (5 x)$

Step 16.2

Combine $\frac{\sin (3 x) \sin (4 x)}{x}$ and $\sec (4 x)$ .

$\frac{\sin (3 x) \sin (4 x) \sec (4 x)}{x} \cot (5 x)$

Step 17

Combine $\frac{\sin (3 x) \sin (4 x) \sec (4 x)}{x}$ and $\cot (5 x)$ .

$\frac{\sin (3 x) \sin (4 x) \sec (4 x) \cot (5 x)}{x}$