Trigonometry Examples

$\csc (x) \cdot \cos (x) = \csc (x)$

Step 1

Simplify the left side.

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Step 1.1

Simplify $\csc (x) \cdot \cos (x)$ .

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Step 1.1.1

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

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

Step 1.1.2

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

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

Step 2

Simplify the right side.

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Step 2.1

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

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

Step 3

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

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

Step 4

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

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Step 4.1

Cancel the common factor.

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

Step 4.2

Rewrite the expression.

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

Step 5

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

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Step 5.1

Cancel the common factor.

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

Step 5.2

Rewrite the expression.

$\cos (x) = 1$

Step 6

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

$x = \arccos (1)$

Step 7

Simplify the right side.

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Step 7.1

The exact value of $\arccos (1)$ is $0$ .

$x = 0$

Step 8

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.

$x = 2 π - 0$

Step 9

Subtract $0$ from $2 π$ .

$x = 2 π$

Step 10

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

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Step 10.1

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

$\frac{2 π}{| b |}$

Step 10.2

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

$\frac{2 π}{| 1 |}$

Step 10.3

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

$\frac{2 π}{1}$

Step 10.4

Divide $2 π$ by $1$ .

$2 π$

Step 11

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

$x = 2 π n, 2 π + 2 π n$ , for any integer $n$

Step 12

Consolidate the answers.

$x = 2 π n$ , for any integer $n$