Trigonometry Examples

$\tan (x) + \cot (x) = 2 \csc (2 x)$

Step 1

Simplify the left side.

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Step 1.1

Simplify each term.

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Step 1.1.1

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

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

Step 1.1.2

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

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

Step 2

Simplify the right side.

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Step 2.1

Simplify $2 \csc (2 x)$ .

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Step 2.1.1

Rewrite $\csc (2 x)$ in terms of sines and cosines.

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

Step 2.1.2

Combine $2$ and $\frac{1}{\sin (2 x)}$ .

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

Step 3

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

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

Step 4

Apply the distributive property.

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

Step 5

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

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Step 5.1

Cancel the common factor.

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

Step 5.2

Rewrite the expression.

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

Step 6

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

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Step 6.1

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

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

Step 6.2

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

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

Step 6.3

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

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

Step 6.4

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

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

Step 6.5

Add $1$ and $1$ .

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

Step 7

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

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

Step 8

Apply the sine double-angle identity.

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

Step 9

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

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Step 9.1

Cancel the common factor.

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

Step 9.2

Rewrite the expression.

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

Step 10

Cancel the common factor of $2$ .

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Step 10.1

Cancel the common factor.

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

Step 10.2

Rewrite the expression.

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

Step 11

Subtract $\frac{1}{\sin (x)}$ from both sides of the equation.

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

Step 12

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

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Step 12.1

Combine the numerators over the common denominator.

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

Step 12.2

Reorder $\cos^{2} (x)$ and $- 1$ .

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

Step 12.3

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

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

Step 12.4

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

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

Step 12.5

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

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

Step 12.6

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

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

Step 12.7

Apply pythagorean identity.

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

Step 12.8

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

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Step 12.8.1

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

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

Step 12.8.2

Cancel the common factors.

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Step 12.8.2.1

Multiply by $1$ .

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

Step 12.8.2.2

Cancel the common factor.

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

Step 12.8.2.3

Rewrite the expression.

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

Step 12.8.2.4

Divide $- \sin (x)$ by $1$ .

$\sin (x) - \sin (x) = 0$

Step 12.9

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

$0 = 0$

Step 13

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

All real numbers

Step 14

The result can be shown in multiple forms.

All real numbers

Interval Notation:

$(- \infty, \infty)$