Trigonometry Examples

$\cos (x) (\tan (x) + \cot (x)) = \csc (x)$

Step 1

Simplify the left side.

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Step 1.1

Simplify $\cos (x) (\tan (x) + \cot (x))$ .

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Step 1.1.1

Simplify each term.

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Step 1.1.1.1

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

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

Step 1.1.1.2

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

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

Step 1.1.2

Simplify terms.

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Step 1.1.2.1

Apply the distributive property.

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

Step 1.1.2.2

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

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Step 1.1.2.2.1

Cancel the common factor.

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

Step 1.1.2.2.2

Rewrite the expression.

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

Step 1.1.3

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

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Step 1.1.3.1

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

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

Step 1.1.3.2

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

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

Step 1.1.3.3

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

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

Step 1.1.3.4

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

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

Step 1.1.3.5

Add $1$ and $1$ .

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

Step 2

Simplify the right side.

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Step 2.1

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

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

Step 3

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

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

Step 4

Apply the distributive property.

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

Step 5

Multiply $\sin (x) \sin (x)$ .

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Step 5.1

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

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

Step 5.2

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

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

Step 5.3

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

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

Step 5.4

Add $1$ and $1$ .

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

Step 6

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

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Step 6.1

Cancel the common factor.

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

Step 6.2

Rewrite the expression.

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

Step 7

Apply pythagorean identity.

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

Step 8

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

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Step 8.1

Cancel the common factor.

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

Step 8.2

Rewrite the expression.

$1 = 1$

Step 9

Since $1 = 1$ , the equation will always be true for any value of $x$ .

All real numbers

Step 10

The result can be shown in multiple forms.

All real numbers

Interval Notation:

$(- \infty, \infty)$