Trigonometry Examples

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

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

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

Step 1.1.2

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

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

Step 1.1.3

Multiply by the reciprocal of the fraction to divide by $\frac{1}{\cos (x)}$ .

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

Step 1.1.4

Write $\cos (x)$ as a fraction with denominator $1$ .

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

Step 1.1.5

Simplify.

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Step 1.1.5.1

Divide $\cos (x)$ by $1$ .

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

Step 1.1.5.2

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

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

Step 1.1.6

Simplify the numerator.

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Step 1.1.6.1

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

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

Step 1.1.6.2

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

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

Step 1.1.6.3

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

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

Step 1.1.6.4

Add $1$ and $1$ .

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

Step 2

Simplify the right side.

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Step 2.1

Simplify $\frac{1 - \cos (x)}{\tan (x)}$ .

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Step 2.1.1

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

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

Step 2.1.2

Multiply the numerator by the reciprocal of the denominator.

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

Step 2.1.3

Apply the distributive property.

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

Step 2.1.4

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

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

Step 2.1.5

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

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Step 2.1.5.1

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

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

Step 2.1.5.2

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

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

Step 2.1.5.3

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

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

Step 2.1.5.4

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

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

Step 2.1.5.5

Add $1$ and $1$ .

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

Step 3

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

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

Step 4

Apply the distributive property.

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

Step 5

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

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Step 5.1

Cancel the common factor.

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

Step 5.2

Rewrite the expression.

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

Step 6

Rewrite using the commutative property of multiplication.

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

Step 7

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

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Step 7.1

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

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

Step 7.2

Cancel the common factor.

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

Step 7.3

Rewrite the expression.

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

Step 8

Apply the distributive property.

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

Step 9

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

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Step 9.1

Cancel the common factor.

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

Step 9.2

Rewrite the expression.

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

Step 10

Rewrite using the commutative property of multiplication.

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

Step 11

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

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Step 11.1

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

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

Step 11.2

Cancel the common factor.

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

Step 11.3

Rewrite the expression.

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

Step 12

Move all terms containing $\cos (x)$ to the left side of the equation.

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Step 12.1

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

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

Step 12.2

Add $\cos^{2} (x)$ to both sides of the equation.

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

Step 12.3

Combine the opposite terms in $\cos (x) - \cos^{2} (x) - \cos (x) + \cos^{2} (x)$ .

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Step 12.3.1

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

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

Step 12.3.2

Add $- \cos^{2} (x)$ and $0$ .

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

Step 12.3.3

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

$0 = 0$

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:

$(- \infty, \infty)$