Trigonometry Examples

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

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.

Tap for more steps...

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.

Tap for more steps...

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.

Tap for more steps...

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.

Tap for more steps...

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.

Tap for more steps...

Step 7.1

Simplify each term.

Tap for more steps...

Step 7.1.1

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

Tap for more steps...

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.

Tap for more steps...

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 $\sin (x)$ .

Tap for more steps...

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.

Tap for more steps...

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.

Tap for more steps...

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:

$(- \infty, \infty)$