Trigonometry Examples

$\sin (x) (\sec (x) \tan (x) + \csc (x) + \cot (x))$

Step 1

Simplify each term.

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Step 1.1

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

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

Step 1.2

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

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

Step 1.3

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

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Step 1.3.1

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

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

Step 1.3.2

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

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

Step 1.3.3

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

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

Step 1.3.4

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

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

Step 1.3.5

Add $1$ and $1$ .

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

Step 1.4

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

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

Step 1.5

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

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

Step 2

Apply the distributive property.

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

Step 3

Simplify.

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Step 3.1

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

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Step 3.1.1

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

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

Step 3.1.2

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

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

Step 3.1.3

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

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

Step 3.1.4

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

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

Step 3.1.5

Add $1$ and $1$ .

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

Step 3.2

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

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Step 3.2.1

Cancel the common factor.

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

Step 3.2.2

Rewrite the expression.

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

Step 3.3

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

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Step 3.3.1

Cancel the common factor.

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

Step 3.3.2

Rewrite the expression.

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

Step 4

Convert from $\frac{\sin^{2} (x)}{\cos^{2} (x)}$ to $\tan^{2} (x)$ .

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

Step 5

Apply pythagorean identity.

$\sec^{2} (x) + \cos (x)$