Trigonometry Examples

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

Step 1

Simplify the left side.

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Step 1.1

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

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Step 1.1.1

Simplify each term.

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Step 1.1.1.1

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

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

Step 1.1.1.2

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

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

Step 1.1.2

Simplify terms.

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Step 1.1.2.1

Apply the distributive property.

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

Step 1.1.2.2

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

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

Step 1.1.2.3

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

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Step 1.1.2.3.1

Cancel the common factor.

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

Step 1.1.2.3.2

Rewrite the expression.

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

Step 2

Simplify the right side.

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Step 2.1

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

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

Step 3

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

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

Step 4

Apply the distributive property.

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

Step 5

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

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Step 5.1

Cancel the common factor.

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

Step 5.2

Rewrite the expression.

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

Step 6

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

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Step 6.1

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

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

Step 6.2

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

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

Step 6.3

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

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

Step 6.4

Add $1$ and $1$ .

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

Step 7

Apply the distributive property.

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

Step 8

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

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Step 8.1

Cancel the common factor.

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

Step 8.2

Rewrite the expression.

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

Step 9

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

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Step 9.1

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

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

Step 9.2

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

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

Step 9.3

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

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

Step 9.4

Add $1$ and $1$ .

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

Step 10

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

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Step 10.1

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

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

Step 10.2

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

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

Step 11

Combine the opposite terms in $\cos (x) + \sin^{2} (x) - \cos (x) - \sin^{2} (x)$ .

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Step 11.1

Subtract $\cos (x)$ from $\cos (x)$ .

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

Step 11.2

Add $0$ and $\sin^{2} (x)$ .

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

Step 11.3

Subtract $\sin^{2} (x)$ from $\sin^{2} (x)$ .

$0 = 0$

Step 12

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

All real numbers

Step 13

The result can be shown in multiple forms.

All real numbers

Interval Notation:

$(- \infty, \infty)$