Trigonometry Examples

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

Step 1

Simplify the left side.

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Step 1.1

Simplify $\frac{\cot^{2} (x)}{\csc (x)}$ .

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Step 1.1.1

Separate fractions.

$\frac{\cot^{2} (x)}{1} \cdot \frac{1}{\csc (x)} = \csc (x) - \sin (x)$

Step 1.1.2

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

$\frac{\cot^{2} (x)}{1} \cdot \frac{1}{\frac{1}{\sin (x)}} = \csc (x) - \sin (x)$

Step 1.1.3

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

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

Step 1.1.4

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

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

Step 1.1.5

Divide $\cot^{2} (x)$ by $1$ .

$\cot^{2} (x) \sin (x) = \csc (x) - \sin (x)$

Step 1.1.6

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

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

Step 1.1.7

Apply the product rule to $\frac{\cos (x)}{\sin (x)}$ .

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

Step 1.1.8

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

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Step 1.1.8.1

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

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

Step 1.1.8.2

Cancel the common factor.

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

Step 1.1.8.3

Rewrite the expression.

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

Step 2

Simplify the right side.

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Step 2.1

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

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

Step 3

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

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

Step 4

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

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Step 4.1

Cancel the common factor.

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

Step 4.2

Rewrite the expression.

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

Step 5

Apply the distributive property.

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

Step 6

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

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Step 6.1

Cancel the common factor.

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

Step 6.2

Rewrite the expression.

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

Step 7

Rewrite using the commutative property of multiplication.

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

Step 8

Multiply $- \sin (x) \sin (x)$ .

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Step 8.1

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

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

Step 8.2

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

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

Step 8.3

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

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

Step 8.4

Add $1$ and $1$ .

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

Step 9

Apply pythagorean identity.

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

Step 10

Since the exponents are equal, the bases of the exponents on both sides of the equation must be equal.

$| \cos (x) | = | \cos (x) |$

Step 11

Solve for $x$ .

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Step 11.1

Rewrite the absolute value equation as four equations without absolute value bars.

$\cos (x) = \cos (x)$

$\cos (x) = - \cos (x)$

$- \cos (x) = \cos (x)$

$- \cos (x) = - \cos (x)$

Step 11.2

After simplifying, there are only two unique equations to be solved.

$\cos (x) = \cos (x)$

$\cos (x) = - \cos (x)$

Step 11.3

Solve $\cos (x) = \cos (x)$ for $x$ .

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Step 11.3.1

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

$x = x$

Step 11.3.2

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

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Step 11.3.2.1

Subtract $x$ from both sides of the equation.

$x - x = 0$

Step 11.3.2.2

Subtract $x$ from $x$ .

$0 = 0$

Step 11.3.3

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

All real numbers

Step 11.4

Solve $\cos (x) = - \cos (x)$ for $x$ .

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Step 11.4.1

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

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Step 11.4.1.1

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

$\cos (x) + \cos (x) = 0$

Step 11.4.1.2

Add $\cos (x)$ and $\cos (x)$ .

$2 \cos (x) = 0$

Step 11.4.2

Divide each term in $2 \cos (x) = 0$ by $2$ and simplify.

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Step 11.4.2.1

Divide each term in $2 \cos (x) = 0$ by $2$ .

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

Step 11.4.2.2

Simplify the left side.

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Step 11.4.2.2.1

Cancel the common factor of $2$ .

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Step 11.4.2.2.1.1

Cancel the common factor.

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

Step 11.4.2.2.1.2

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

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

Step 11.4.2.3

Simplify the right side.

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Step 11.4.2.3.1

Divide $0$ by $2$ .

$\cos (x) = 0$

Step 11.4.3

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

$x = \arccos (0)$

Step 11.4.4

Simplify the right side.

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Step 11.4.4.1

The exact value of $\arccos (0)$ is $\frac{π}{2}$ .

$x = \frac{π}{2}$

Step 11.4.5

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.

$x = 2 π - \frac{π}{2}$

Step 11.4.6

Simplify $2 π - \frac{π}{2}$ .

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Step 11.4.6.1

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

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

Step 11.4.6.2

Combine fractions.

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Step 11.4.6.2.1

Combine $2 π$ and $\frac{2}{2}$ .

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

Step 11.4.6.2.2

Combine the numerators over the common denominator.

$x = \frac{2 π \cdot 2 - π}{2}$

Step 11.4.6.3

Simplify the numerator.

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Step 11.4.6.3.1

Multiply $2$ by $2$ .

$x = \frac{4 π - π}{2}$

Step 11.4.6.3.2

Subtract $π$ from $4 π$ .

$x = \frac{3 π}{2}$

Step 11.4.7

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

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Step 11.4.7.1

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

$\frac{2 π}{| b |}$

Step 11.4.7.2

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

$\frac{2 π}{| 1 |}$

Step 11.4.7.3

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

$\frac{2 π}{1}$

Step 11.4.7.4

Divide $2 π$ by $1$ .

$2 π$

Step 11.4.8

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

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

Step 12

Consolidate the answers.

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