Trigonometry Examples

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

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

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

Step 1.1.2

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

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

Step 2

Simplify the right side.

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Step 2.1

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

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

Step 3

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

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

Step 4

Apply the distributive property.

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

Step 5

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

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Step 5.1

Cancel the common factor.

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

Step 5.2

Rewrite the expression.

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

Step 6

Rewrite using the commutative property of multiplication.

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

Step 7

Simplify each term.

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Step 7.1

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

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

Step 7.2

Use the double-angle identity to transform $\cos (2 x)$ to $2 \cos^{2} (x) - 1$ .

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

Step 7.3

Apply the sine double-angle identity.

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

Step 7.4

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

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Step 7.4.1

Cancel the common factor.

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

Step 7.4.2

Rewrite the expression.

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

Step 7.5

Apply the cosine double-angle identity.

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

Step 8

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

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Step 8.1

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

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

Step 8.2

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

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

Step 8.3

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

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

Step 8.4

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

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

Step 8.5

Add $1$ and $1$ .

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

Step 9

Subtract $\frac{\sin^{2} (x)}{\cos (x)}$ from both sides of the equation.

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

Step 10

Simplify each term.

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Step 10.1

Separate fractions.

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

Step 10.2

Rewrite $\frac{\cos (2 x)}{\cos (x)}$ as a product.

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

Step 10.3

Write $\cos (2 x)$ as a fraction with denominator $1$ .

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

Step 10.4

Simplify.

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Step 10.4.1

Divide $\cos (2 x)$ by $1$ .

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

Step 10.4.2

Convert from $\frac{1}{\cos (x)}$ to $\sec (x)$ .

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

Step 10.5

Multiply $\frac{1}{2} (\cos (2 x) \sec (x))$ .

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Step 10.5.1

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

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

Step 10.5.2

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

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

Step 10.6

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

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

Step 10.7

Separate fractions.

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

Step 10.8

Convert from $\frac{\sin (x)}{\cos (x)}$ to $\tan (x)$ .

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

Step 10.9

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

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

Step 11

Simplify the left side.

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Step 11.1

Simplify each term.

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Step 11.1.1

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

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

Step 11.1.2

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

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

Step 11.1.3

Multiply the numerator by the reciprocal of the denominator.

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

Step 11.1.4

Multiply $\frac{\cos (2 x)}{\cos (x)}$ by $\frac{1}{2}$ .

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

Step 11.1.5

Move $2$ to the left of $\cos (x)$ .

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

Step 11.1.6

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

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

Step 11.1.7

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

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Step 11.1.7.1

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

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

Step 11.1.7.2

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

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

Step 11.1.7.3

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

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

Step 11.1.7.4

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

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

Step 11.1.7.5

Add $1$ and $1$ .

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

Step 12

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

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

Step 13

Apply the distributive property.

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

Step 14

Simplify.

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Step 14.1

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

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

Step 14.2

Rewrite using the commutative property of multiplication.

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

Step 14.3

Rewrite using the commutative property of multiplication.

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

Step 15

Simplify each term.

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Step 15.1

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

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Step 15.1.1

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

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

Step 15.1.2

Factor $\cos (x)$ out of $2 \cos (x)$ .

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

Step 15.1.3

Cancel the common factor.

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

Step 15.1.4

Rewrite the expression.

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

Step 15.2

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

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Step 15.2.1

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

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

Step 15.2.2

Cancel the common factor.

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

Step 15.2.3

Rewrite the expression.

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

Step 16

Multiply $\cos (x)$ by $0$ .

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

Step 17

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

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

Step 18

Use the double-angle identity to transform $\cos (2 x)$ to $1 - 2 \sin^{2} (x)$ .

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

Step 19

Simplify the left side.

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Step 19.1

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

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Step 19.1.1

Apply pythagorean identity.

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

Step 19.1.2

To write $\cos (x)$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

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

Step 19.1.3

Simplify terms.

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Step 19.1.3.1

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

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

Step 19.1.3.2

Combine the numerators over the common denominator.

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

Step 19.1.4

Simplify the numerator.

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Step 19.1.4.1

Move $2$ to the left of $\cos (x)$ .

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

Step 19.1.4.2

Apply the cosine double-angle identity.

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

Step 19.1.4.3

Simplify each term.

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Step 19.1.4.3.1

Use the double-angle identity to transform $\cos (2 x)$ to $2 \cos^{2} (x) - 1$ .

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

Step 19.1.4.3.2

Apply the distributive property.

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

Step 19.1.4.3.3

Multiply $2$ by $- 1$ .

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

Step 19.1.4.3.4

Multiply $- 1$ by $- 1$ .

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

Step 19.1.5

To write $- \sin^{2} (x)$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

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

Step 19.1.6

Simplify terms.

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Step 19.1.6.1

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

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

Step 19.1.6.2

Combine the numerators over the common denominator.

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

Step 19.1.7

Simplify the numerator.

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Step 19.1.7.1

Multiply $2$ by $- 1$ .

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

Step 19.1.7.2

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

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

Step 19.1.7.3

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

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

Step 19.1.7.4

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

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

Step 19.1.7.5

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

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

Step 19.1.7.6

Rearrange terms.

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

Step 19.1.7.7

Apply pythagorean identity.

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

Step 19.1.7.8

Multiply $- 2$ by $1$ .

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

Step 19.1.7.9

Add $- 2$ and $1$ .

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

Step 20

Solve the equation for $x$ .

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Step 20.1

Set the numerator equal to zero.

$2 \cos (x) - 1 = 0$

Step 20.2

Solve the equation for $x$ .

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Step 20.2.1

Add $1$ to both sides of the equation.

$2 \cos (x) = 1$

Step 20.2.2

Divide each term in $2 \cos (x) = 1$ by $2$ and simplify.

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Step 20.2.2.1

Divide each term in $2 \cos (x) = 1$ by $2$ .

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

Step 20.2.2.2

Simplify the left side.

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Step 20.2.2.2.1

Cancel the common factor of $2$ .

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Step 20.2.2.2.1.1

Cancel the common factor.

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

Step 20.2.2.2.1.2

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

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

Step 20.2.3

Take the inverse cosine of both sides of the equation to extract $x$ from inside the cosine.

$x = \arccos (\frac{1}{2})$

Step 20.2.4

Simplify the right side.

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Step 20.2.4.1

The exact value of $\arccos (\frac{1}{2})$ is $\frac{π}{3}$ .

$x = \frac{π}{3}$

Step 20.2.5

The cosine function is positive in the first and fourth quadrants. To find the second solution, subtract the reference angle from $2 π$ to find the solution in the fourth quadrant.

$x = 2 π - \frac{π}{3}$

Step 20.2.6

Simplify $2 π - \frac{π}{3}$ .

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Step 20.2.6.1

To write $2 π$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$x = 2 π \cdot \frac{3}{3} - \frac{π}{3}$

Step 20.2.6.2

Combine fractions.

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Step 20.2.6.2.1

Combine $2 π$ and $\frac{3}{3}$ .

$x = \frac{2 π \cdot 3}{3} - \frac{π}{3}$

Step 20.2.6.2.2

Combine the numerators over the common denominator.

$x = \frac{2 π \cdot 3 - π}{3}$

Step 20.2.6.3

Simplify the numerator.

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Step 20.2.6.3.1

Multiply $3$ by $2$ .

$x = \frac{6 π - π}{3}$

Step 20.2.6.3.2

Subtract $π$ from $6 π$ .

$x = \frac{5 π}{3}$

Step 20.2.7

Find the period of $\cos (x)$ .

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Step 20.2.7.1

The period of the function can be calculated using $\frac{2 π}{| b |}$ .

$\frac{2 π}{| b |}$

Step 20.2.7.2

Replace $b$ with $1$ in the formula for period.

$\frac{2 π}{| 1 |}$

Step 20.2.7.3

The absolute value is the distance between a number and zero. The distance between $0$ and $1$ is $1$ .

$\frac{2 π}{1}$

Step 20.2.7.4

Divide $2 π$ by $1$ .

$2 π$

Step 20.2.8

The period of the $\cos (x)$ function is $2 π$ so values will repeat every $2 π$ radians in both directions.

$x = \frac{π}{3} + 2 π n, \frac{5 π}{3} + 2 π n$ , for any integer $n$