Trigonometry Examples

$\sin (x) (\frac{\cos (x) \csc (x) + \cot (x)}{\csc (x) \cot (x)})$

Step 1

Simplify the numerator.

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Step 1.1

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

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

Step 1.2

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

$\sin (x) \frac{\frac{\cos (x)}{\sin (x)} + \cot (x)}{\csc (x) \cot (x)}$

Step 1.3

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

$\sin (x) \frac{\frac{\cos (x)}{\sin (x)} + \frac{\cos (x)}{\sin (x)}}{\csc (x) \cot (x)}$

Step 1.4

Add $\frac{\cos (x)}{\sin (x)}$ and $\frac{\cos (x)}{\sin (x)}$ .

$\sin (x) \frac{2 \frac{\cos (x)}{\sin (x)}}{\csc (x) \cot (x)}$

Step 2

Simplify the denominator.

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Step 2.1

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

$\sin (x) \frac{2 \frac{\cos (x)}{\sin (x)}}{\frac{1}{\sin (x)} \cot (x)}$

Step 2.2

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

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

Step 3

Combine fractions.

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Step 3.1

Combine $2$ and $\frac{\cos (x)}{\sin (x)}$ .

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

Step 3.2

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

$\sin (x) \frac{\frac{2 \cos (x)}{\sin (x)}}{\frac{\cos (x)}{\sin (x) \sin (x)}}$

Step 4

Simplify the denominator.

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Step 4.1

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

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

Step 4.2

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

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

Step 4.3

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

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

Step 4.4

Add $1$ and $1$ .

$\sin (x) \frac{\frac{2 \cos (x)}{\sin (x)}}{\frac{\cos (x)}{\sin^{2} (x)}}$

Step 5

Multiply the numerator by the reciprocal of the denominator.

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

Step 6

Combine.

$\sin (x) \frac{2 \cos (x) \sin^{2} (x)}{\sin (x) \cos (x)}$

Step 7

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

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Step 7.1

Factor $\sin (x)$ out of $\sin (x) \cos (x)$ .

$\sin (x) \frac{2 \cos (x) \sin^{2} (x)}{\sin (x) (\cos (x))}$

Step 7.2

Cancel the common factor.

$\sin (x) \frac{2 \cos (x) \sin^{2} (x)}{\sin (x) \cos (x)}$

Step 7.3

Rewrite the expression.

$\frac{2 \cos (x) \sin^{2} (x)}{\cos (x)}$

Step 8

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

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Step 8.1

Cancel the common factor.

$\frac{2 \cos (x) \sin^{2} (x)}{\cos (x)}$

Step 8.2

Divide $2 \sin^{2} (x)$ by $1$ .

$2 \sin^{2} (x)$