Trigonometry Examples

$\csc (x) \cdot \sec (x) = \cot (x) + \tan (x)$

Step 1

Simplify the left side.

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Step 1.1

Simplify $\csc (x) \cdot \sec (x)$ .

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Step 1.1.1

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

$\frac{1}{\sin (x)} \cdot \sec (x) = \cot (x) + \tan (x)$

Step 1.1.2

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

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

Step 1.1.3

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

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

Step 2

Simplify the right side.

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Step 2.1

Simplify each term.

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Step 2.1.1

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

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

Step 2.1.2

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

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

Step 3

Multiply both sides of the equation by $\sin (x)$ .

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

Step 4

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

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Step 4.1

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

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

Step 4.2

Cancel the common factor.

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

Step 4.3

Rewrite the expression.

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

Step 5

Apply the distributive property.

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

Step 6

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

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Step 6.1

Cancel the common factor.

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

Step 6.2

Rewrite the expression.

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

Step 7

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

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Step 7.1

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

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

Step 7.2

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

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

Step 7.3

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

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

Step 7.4

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

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

Step 7.5

Add $1$ and $1$ .

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

Step 8

Move all the expressions to the left side of the equation.

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Step 8.1

Subtract $\cos (x)$ from both sides of the equation.

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

Step 8.2

Subtract $\frac{\sin^{2} (x)}{\cos (x)}$ from both sides of the equation.

$\frac{1}{\cos (x)} - \cos (x) - \frac{\sin^{2} (x)}{\cos (x)} = 0$

Step 9

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

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Step 9.1

Combine the numerators over the common denominator.

$- \cos (x) + \frac{1 - \sin^{2} (x)}{\cos (x)} = 0$

Step 9.2

Apply pythagorean identity.

$- \cos (x) + \frac{\cos^{2} (x)}{\cos (x)} = 0$

Step 9.3

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

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Step 9.3.1

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

$- \cos (x) + \frac{\cos (x) \cos (x)}{\cos (x)} = 0$

Step 9.3.2

Cancel the common factors.

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Step 9.3.2.1

Multiply by $1$ .

$- \cos (x) + \frac{\cos (x) \cos (x)}{\cos (x) \cdot 1} = 0$

Step 9.3.2.2

Cancel the common factor.

$- \cos (x) + \frac{\cos (x) \cos (x)}{\cos (x) \cdot 1} = 0$

Step 9.3.2.3

Rewrite the expression.

$- \cos (x) + \frac{\cos (x)}{1} = 0$

Step 9.3.2.4

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

$- \cos (x) + \cos (x) = 0$

Step 9.4

Add $- \cos (x)$ and $\cos (x)$ .

$0 = 0$

Step 10

Since $0 = 0$ , the equation will always be true for any value of $x$ .

All real numbers

Step 11

The result can be shown in multiple forms.

All real numbers

Interval Notation:

$(- \infty, \infty)$