Trigonometry Examples

$\cos (x) (\tan (x)) + \cot (x) = \csc (x)$

Step 1

Simplify the left side.

Tap for more steps...

Step 1.1

Simplify each term.

Tap for more steps...

Step 1.1.1

Rewrite in terms of sines and cosines, then cancel the common factors.

Tap for more steps...

Step 1.1.1.1

Reorder $\cos (x)$ and $\tan (x)$ .

$\tan (x) \cos (x) + \cot (x) = \csc (x)$

Step 1.1.1.2

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

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

Step 1.1.1.3

Cancel the common factors.

$\sin (x) + \cot (x) = \csc (x)$

Step 1.1.2

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

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

Step 2

Simplify the right side.

Tap for more steps...

Step 2.1

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

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

Step 3

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

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

Step 4

Apply the distributive property.

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

Step 5

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

Tap for more steps...

Step 5.1

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

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

Step 5.2

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

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

Step 5.3

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

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

Step 5.4

Add $1$ and $1$ .

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

Step 6

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

Tap for more steps...

Step 6.1

Cancel the common factor.

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

Step 6.2

Rewrite the expression.

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

Step 7

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

Tap for more steps...

Step 7.1

Cancel the common factor.

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

Step 7.2

Rewrite the expression.

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

Step 8

Subtract $1$ from both sides of the equation.

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

Step 9

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

Tap for more steps...

Step 9.1

Move $- 1$ .

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

Step 9.2

Reorder $\sin^{2} (x)$ and $- 1$ .

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

Step 9.3

Rewrite $- 1$ as $- 1 (1)$ .

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

Step 9.4

Factor $- 1$ out of $\sin^{2} (x)$ .

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

Step 9.5

Factor $- 1$ out of $- 1 (1) - 1 (- \sin^{2} (x))$ .

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

Step 9.6

Rewrite $- 1 (1 - \sin^{2} (x))$ as $- (1 - \sin^{2} (x))$ .

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

Step 9.7

Apply pythagorean identity.

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

Step 10

Solve for $x$ .

Tap for more steps...

Step 10.1

Factor the left side of the equation.

Tap for more steps...

Step 10.1.1

Let $u = \cos (x)$ . Substitute $u$ for all occurrences of $\cos (x)$ .

$- u^{2} + u = 0$

Step 10.1.2

Factor $u$ out of $- u^{2} + u$ .

Tap for more steps...

Step 10.1.2.1

Factor $u$ out of $- u^{2}$ .

$u (- u) + u = 0$

Step 10.1.2.2

Raise $u$ to the power of $1$ .

$u (- u) + u = 0$

Step 10.1.2.3

Factor $u$ out of $u^{1}$ .

$u (- u) + u \cdot 1 = 0$

Step 10.1.2.4

Factor $u$ out of $u (- u) + u \cdot 1$ .

$u (- u + 1) = 0$

Step 10.1.3

Replace all occurrences of $u$ with $\cos (x)$ .

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

Step 10.2

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$\cos (x) = 0$

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

Step 10.3

Set $\cos (x)$ equal to $0$ and solve for $x$ .

Tap for more steps...

Step 10.3.1

Set $\cos (x)$ equal to $0$ .

$\cos (x) = 0$

Step 10.3.2

Solve $\cos (x) = 0$ for $x$ .

Tap for more steps...

Step 10.3.2.1

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

$x = \arccos (0)$

Step 10.3.2.2

Simplify the right side.

Tap for more steps...

Step 10.3.2.2.1

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

$x = \frac{π}{2}$

Step 10.3.2.3

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 10.3.2.4

Simplify $2 π - \frac{π}{2}$ .

Tap for more steps...

Step 10.3.2.4.1

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 10.3.2.4.2

Combine fractions.

Tap for more steps...

Step 10.3.2.4.2.1

Combine $2 π$ and $\frac{2}{2}$ .

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

Step 10.3.2.4.2.2

Combine the numerators over the common denominator.

$x = \frac{2 π \cdot 2 - π}{2}$

Step 10.3.2.4.3

Simplify the numerator.

Tap for more steps...

Step 10.3.2.4.3.1

Multiply $2$ by $2$ .

$x = \frac{4 π - π}{2}$

Step 10.3.2.4.3.2

Subtract $π$ from $4 π$ .

$x = \frac{3 π}{2}$

Step 10.3.2.5

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

Tap for more steps...

Step 10.3.2.5.1

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

$\frac{2 π}{| b |}$

Step 10.3.2.5.2

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

$\frac{2 π}{| 1 |}$

Step 10.3.2.5.3

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

$\frac{2 π}{1}$

Step 10.3.2.5.4

Divide $2 π$ by $1$ .

$2 π$

Step 10.3.2.6

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 10.4

Set $- \cos (x) + 1$ equal to $0$ and solve for $x$ .

Tap for more steps...

Step 10.4.1

Set $- \cos (x) + 1$ equal to $0$ .

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

Step 10.4.2

Solve $- \cos (x) + 1 = 0$ for $x$ .

Tap for more steps...

Step 10.4.2.1

Subtract $1$ from both sides of the equation.

$- \cos (x) = - 1$

Step 10.4.2.2

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

Tap for more steps...

Step 10.4.2.2.1

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

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

Step 10.4.2.2.2

Simplify the left side.

Tap for more steps...

Step 10.4.2.2.2.1

Dividing two negative values results in a positive value.

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

Step 10.4.2.2.2.2

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

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

Step 10.4.2.2.3

Simplify the right side.

Tap for more steps...

Step 10.4.2.2.3.1

Divide $- 1$ by $- 1$ .

$\cos (x) = 1$

Step 10.4.2.3

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

$x = \arccos (1)$

Step 10.4.2.4

Simplify the right side.

Tap for more steps...

Step 10.4.2.4.1

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

$x = 0$

Step 10.4.2.5

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.

$x = 2 π - 0$

Step 10.4.2.6

Subtract $0$ from $2 π$ .

$x = 2 π$

Step 10.4.2.7

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

Tap for more steps...

Step 10.4.2.7.1

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

$\frac{2 π}{| b |}$

Step 10.4.2.7.2

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

$\frac{2 π}{| 1 |}$

Step 10.4.2.7.3

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

$\frac{2 π}{1}$

Step 10.4.2.7.4

Divide $2 π$ by $1$ .

$2 π$

Step 10.4.2.8

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 10.5

The final solution is all the values that make $\cos (x) (- \cos (x) + 1) = 0$ true.

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

Step 11

Consolidate the answers.

Tap for more steps...

Step 11.1

Consolidate $\frac{π}{2} + 2 π n$ and $\frac{3 π}{2} + 2 π n$ to $\frac{π}{2} + π n$ .

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

Step 11.2

Consolidate $2 π n$ and $2 π + 2 π n$ to $2 π n$ .

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

Step 12

Exclude the solutions that do not make $\cos (x) (\tan (x)) + \cot (x) = \csc (x)$ true.

$x = 1.57079632, 4.71238898, 6.2831853, 7.85398163, 10.99557428, 12.56637061, 14.13716694, 17.27875959, 18.84955592, 20.42035224, 23.5619449, 25.13274122, 31.41592653, 37.69911184, 43.98229715$ , for any integer $n$