Trigonometry Examples

(cot (x) - csc (x)) (cos (x) + 1) = - sin (x)

$(\cot (x) - \csc (x)) (\cos (x) + 1) = - \sin (x)$

Step 1

Divide each term in the equation by

$\cos (x)$ .

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

Step 2

Simplify the numerator.

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Step 2.1

Rewrite

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

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

Step 2.2

Rewrite

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

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

Step 3

Simplify the numerator.

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Step 3.1

Convert from

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

$\csc (x)$ .

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

Step 3.2

Convert from

$\frac{\cos (x)}{\sin (x)}$ to

$\cot (x)$ .

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

Step 4

Replace

$\cos (x)$ with an equivalent expression in the numerator.

$(\cot (x) - \csc (x)) (\cos (x) + 1) \cdot \sec (x) = \frac{- \sin (x)}{\cos (x)}$

Step 5

Simplify each term.

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Step 5.1

Rewrite

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

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

Step 5.2

Rewrite

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

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

Step 6

Expand

$(\frac{\cos (x)}{\sin (x)} - \frac{1}{\sin (x)}) (\cos (x) + 1)$ using the FOIL Method.

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Step 6.1

Apply the distributive property.

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

Step 6.2

Apply the distributive property.

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

Step 6.3

Apply the distributive property.

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

Step 7

Simplify and combine like terms.

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Step 7.1

Simplify each term.

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Step 7.1.1

Multiply

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

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Step 7.1.1.1

Combine

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

$\cos (x)$ .

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

Step 7.1.1.2

Raise

$\cos (x)$ to the power of

$1$ .

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

Step 7.1.1.3

Raise

$\cos (x)$ to the power of

$1$ .

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

Step 7.1.1.4

Use the power rule

$a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Step 7.1.1.5

Add

$1$ and

$1$ .

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

Step 7.1.2

Multiply

$\frac{\cos (x)}{\sin (x)}$ by

$1$ .

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

Step 7.1.3

Combine

$\cos (x)$ and

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

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

Step 7.1.4

Multiply

$- 1$ by

$1$ .

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

Step 7.2

Subtract

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

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

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

Step 7.3

Add

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

$0$ .

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

Step 8

Simplify terms.

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Step 8.1

Combine the numerators over the common denominator.

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

Step 8.2

Reorder

$\cos^{2} (x)$ and

$- 1$ .

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

Step 8.3

Rewrite

$- 1$ as

$- 1 (1)$ .

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

Step 8.4

Factor

$- 1$ out of

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

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

Step 8.5

Factor

$- 1$ out of

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

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

Step 8.6

Rewrite

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

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

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

Step 9

Apply pythagorean identity.

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

Step 10

Cancel the common factor of

$\sin^{2} (x)$ and

$\sin (x)$ .

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Step 10.1

Factor

$\sin (x)$ out of

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

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

Step 10.2

Cancel the common factors.

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Step 10.2.1

Multiply by

$1$ .

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

Step 10.2.2

Cancel the common factor.

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

Step 10.2.3

Rewrite the expression.

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

Step 10.2.4

Divide

$- \sin (x)$ by

$1$ .

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

Step 11

Rewrite

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

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

Step 12

Combine

$\frac{1}{\cos (x)}$ and

$\sin (x)$ .

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

Step 13

Convert from

$\frac{\sin (x)}{\cos (x)}$ to

$\tan (x)$ .

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

Step 14

Separate fractions.

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

Step 15

Convert from

$\frac{\sin (x)}{\cos (x)}$ to

$\tan (x)$ .

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

Step 16

Divide

$- 1$ by

$1$ .

$- \tan (x) = - \tan (x)$

Step 17

Move all terms containing

$\tan (x)$ to the left side of the equation.

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Step 17.1

Add

$\tan (x)$ to both sides of the equation.

$- \tan (x) + \tan (x) = 0$

Step 17.2

Add

$- \tan (x)$ and

$\tan (x)$ .

$0 = 0$

Step 18

Since

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

$x$ .

All real numbers

Step 19

The result can be shown in multiple forms.

All real numbers

Interval Notation:

$(- \infty, \infty)$