Trigonometry Examples

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

Step 1

Simplify the left side.

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Step 1.1

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

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

Step 2

Simplify the right side.

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Step 2.1

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

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

Step 3

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

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

Step 4

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

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Step 4.1

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

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

Step 4.2

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

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

Step 4.3

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

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

Step 4.4

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

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

Step 4.5

Add $1$ and $1$ .

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

Step 5

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

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Step 5.1

Cancel the common factor.

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

Step 5.2

Rewrite the expression.

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

Step 6

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

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

Step 7

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

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Step 7.1

Convert from $\frac{1}{\sin (x)}$ to $\csc (x)$ .

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

Step 7.2

To write $- \sin (x)$ as a fraction with a common denominator, multiply by $\frac{\csc (x) - \sin (x)}{\csc (x) - \sin (x)}$ .

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

Step 7.3

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

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

Step 7.4

Combine the numerators over the common denominator.

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

Step 7.5

Simplify the numerator.

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Step 7.5.1

Apply the distributive property.

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

Step 7.5.2

Rewrite in terms of sines and cosines, then cancel the common factors.

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Step 7.5.2.1

Add parentheses.

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

Step 7.5.2.2

Reorder $\sin (x)$ and $\csc (x)$ .

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

Step 7.5.2.3

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

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

Step 7.5.2.4

Cancel the common factors.

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

Step 7.5.3

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

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Step 7.5.3.1

Multiply $- 1$ by $- 1$ .

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

Step 7.5.3.2

Multiply $\sin (x)$ by $1$ .

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

Step 7.5.3.3

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

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

Step 7.5.3.4

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

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

Step 7.5.3.5

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

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

Step 7.5.3.6

Add $1$ and $1$ .

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

Step 7.5.4

Multiply $- 1$ by $1$ .

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

Step 7.5.5

Rewrite $- 1$ as $- 1 (1)$ .

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

Step 7.5.6

Factor $- 1$ out of $\sin^{2} (x)$ .

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

Step 7.5.7

Factor $- 1$ out of $- 1 (1) - 1 (- \sin^{2} (x))$ .

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

Step 7.5.8

Rewrite $- 1 (1 - \sin^{2} (x))$ as $- (1 - \sin^{2} (x))$ .

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

Step 7.5.9

Apply pythagorean identity.

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

Step 7.5.10

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

$\frac{0}{\csc (x) - \sin (x)} = 0$

Step 7.6

Divide $0$ by $\csc (x) - \sin (x)$ .

$0 = 0$

Step 8

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

All real numbers

Step 9

The result can be shown in multiple forms.

All real numbers

Interval Notation:

$(- \infty, \infty)$