Trigonometry Examples

$\frac{\tan (x) + \cot (y)}{\tan (x) \cdot \cot (y)} = \tan (y) + \cot (x)$

Step 1

Multiply both sides by $\tan (x) \cdot \cot (y)$ .

$\frac{\tan (x) + \cot (y)}{\tan (x) \cdot \cot (y)} (\tan (x) \cdot \cot (y)) = (\tan (y) + \cot (x)) (\tan (x) \cdot \cot (y))$

Step 2

Simplify.

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Step 2.1

Simplify the left side.

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Step 2.1.1

Cancel the common factor of $\tan (x) \cdot \cot (y)$ .

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Step 2.1.1.1

Cancel the common factor.

$\frac{\tan (x) + \cot (y)}{\tan (x) \cdot \cot (y)} (\tan (x) \cdot \cot (y)) = (\tan (y) + \cot (x)) (\tan (x) \cdot \cot (y))$

Step 2.1.1.2

Rewrite the expression.

$\tan (x) + \cot (y) = (\tan (y) + \cot (x)) (\tan (x) \cdot \cot (y))$

Step 2.2

Simplify the right side.

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Step 2.2.1

Simplify $(\tan (y) + \cot (x)) (\tan (x) \cdot \cot (y))$ .

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Step 2.2.1.1

Apply the distributive property.

$\tan (x) + \cot (y) = \tan (y) (\tan (x) \cot (y)) + \cot (x) (\tan (x) \cot (y))$

Step 2.2.1.2

Simplify each term.

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Step 2.2.1.2.1

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

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Step 2.2.1.2.1.1

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

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

Step 2.2.1.2.1.2

Cancel the common factors.

$\tan (x) + \cot (y) = \tan (y) \tan (x) \cot (y) + 1 \cot (y)$

Step 2.2.1.2.2

Multiply $\cot (y)$ by $1$ .

$\tan (x) + \cot (y) = \tan (y) \tan (x) \cot (y) + \cot (y)$

Step 3

Solve for $x$ .

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Step 3.1

Subtract $\tan (y) \tan (x) \cot (y)$ from both sides of the equation.

$\tan (x) + \cot (y) - \tan (y) \tan (x) \cot (y) = \cot (y)$

Step 3.2

Simplify the left side.

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Step 3.2.1

Simplify $\tan (x) + \cot (y) - \tan (y) \tan (x) \cot (y)$ .

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Step 3.2.1.1

Simplify each term.

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Step 3.2.1.1.1

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

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

Step 3.2.1.1.2

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

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

Step 3.2.1.1.3

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

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

Step 3.2.1.1.4

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

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

Step 3.2.1.1.5

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

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

Step 3.2.1.1.6

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

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

Step 3.2.1.1.7

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

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Step 3.2.1.1.7.1

Move the leading negative in $- \frac{\sin (x) \sin (y)}{\cos (x) \cos (y)}$ into the numerator.

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

Step 3.2.1.1.7.2

Factor $\sin (y)$ out of $- \sin (x) \sin (y)$ .

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

Step 3.2.1.1.7.3

Cancel the common factor.

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

Step 3.2.1.1.7.4

Rewrite the expression.

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

Step 3.2.1.1.8

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

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Step 3.2.1.1.8.1

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

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

Step 3.2.1.1.8.2

Cancel the common factor.

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

Step 3.2.1.1.8.3

Rewrite the expression.

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

Step 3.2.1.1.9

Move the negative in front of the fraction.

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

Step 3.2.1.2

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

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Step 3.2.1.2.1

Subtract $\frac{\sin (x)}{\cos (x)}$ from $\frac{\sin (x)}{\cos (x)}$ .

$0 + \frac{\cos (y)}{\sin (y)} = \cot (y)$

Step 3.2.1.2.2

Add $0$ and $\frac{\cos (y)}{\sin (y)}$ .

$\frac{\cos (y)}{\sin (y)} = \cot (y)$

Step 3.3

Simplify the right side.

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Step 3.3.1

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

$\frac{\cos (y)}{\sin (y)} = \frac{\cos (y)}{\sin (y)}$

Step 3.4

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

$\sin (y) \frac{\cos (y)}{\sin (y)} = \sin (y) \frac{\cos (y)}{\sin (y)}$

Step 3.5

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

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Step 3.5.1

Cancel the common factor.

$\sin (y) \frac{\cos (y)}{\sin (y)} = \sin (y) \frac{\cos (y)}{\sin (y)}$

Step 3.5.2

Rewrite the expression.

$\cos (y) = \sin (y) \frac{\cos (y)}{\sin (y)}$

Step 3.6

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

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Step 3.6.1

Cancel the common factor.

$\cos (y) = \sin (y) \frac{\cos (y)}{\sin (y)}$

Step 3.6.2

Rewrite the expression.

$\cos (y) = \cos (y)$

Step 3.7

For the two functions to be equal, the arguments of each must be equal.

$y = y$

Step 3.8

Move all terms containing $y$ to the left side of the equation.

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Step 3.8.1

Subtract $y$ from both sides of the equation.

$y - y = 0$

Step 3.8.2

Subtract $y$ from $y$ .

$0 = 0$

Step 3.9

Since $0 = 0$ , the equation will always be true.

Always true

Step 4

The result can be shown in multiple forms.

Always true

Interval Notation:

$(- \infty, \infty)$