Trigonometry Examples

$\frac{\csc (x) - \sin (x)}{\cot (x)}$

Step 1

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

$\frac{\frac{1}{\sin (x)} - \sin (x)}{\cot (x)}$

Step 2

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

$\frac{\frac{1}{\sin (x)} - \sin (x)}{\frac{\cos (x)}{\sin (x)}}$

Step 3

Multiply the numerator by the reciprocal of the denominator.

$(\frac{1}{\sin (x)} - \sin (x)) \frac{\sin (x)}{\cos (x)}$

Step 4

Apply the distributive property.

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

Step 5

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

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Step 5.1

Cancel the common factor.

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

Step 5.2

Rewrite the expression.

$\frac{1}{\cos (x)} - \sin (x) \frac{\sin (x)}{\cos (x)}$

Step 6

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

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Step 6.1

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

$\frac{1}{\cos (x)} - \frac{\sin (x) \sin (x)}{\cos (x)}$

Step 6.2

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

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

Step 6.3

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

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

Step 6.4

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

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

Step 6.5

Add $1$ and $1$ .

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

Step 7

Combine the numerators over the common denominator.

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

Step 8

Apply pythagorean identity.

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

Step 9

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

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Step 9.1

Factor $\cos (x)$ out of $\cos^{2} (x)$ .

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

Step 9.2

Cancel the common factors.

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Step 9.2.1

Multiply by $1$ .

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

Step 9.2.2

Cancel the common factor.

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

Step 9.2.3

Rewrite the expression.

$\frac{\cos (x)}{1}$

Step 9.2.4

Divide $\cos (x)$ by $1$ .

$\cos (x)$