Trigonometry Examples

$\cos (x) (\csc (x) - \sec (x) - \cot (x))$

Step 1

Simplify each term.

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Step 1.1

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

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

Step 1.2

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

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

Step 1.3

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

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

Step 2

Apply the distributive property.

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

Step 3

Simplify.

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Step 3.1

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

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

Step 3.2

Rewrite using the commutative property of multiplication.

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

Step 3.3

Rewrite using the commutative property of multiplication.

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

Step 4

Simplify each term.

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Step 4.1

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

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Step 4.1.1

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

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

Step 4.1.2

Cancel the common factor.

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

Step 4.1.3

Rewrite the expression.

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

Step 4.2

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

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Step 4.2.1

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

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

Step 4.2.2

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

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

Step 4.2.3

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

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

Step 4.2.4

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

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

Step 4.2.5

Add $1$ and $1$ .

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

Step 5

Simplify each term.

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Step 5.1

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

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

Step 5.2

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

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

Step 5.3

Separate fractions.

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

Step 5.4

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

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

Step 5.5

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

$\cot (x) - 1 - \cos (x) \cot (x)$