Trigonometry Examples

$\csc (x) (\csc (x) + \sin (- x))$

Step 1

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

$\frac{1}{\sin (x)} (\csc (x) + \sin (- x))$

Step 2

Simplify each term.

Tap for more steps...

Step 2.1

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

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

Step 2.2

Since $\sin (- x)$ is an odd function, rewrite $\sin (- x)$ as $- \sin (x)$ .

$\frac{1}{\sin (x)} (\frac{1}{\sin (x)} - \sin (x))$

Step 3

Apply the distributive property.

$\frac{1}{\sin (x)} \cdot \frac{1}{\sin (x)} + \frac{1}{\sin (x)} (- \sin (x))$

Step 4

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

Tap for more steps...

Step 4.1

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

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

Step 4.2

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

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

Step 4.3

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

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

Step 4.4

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

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

Step 4.5

Add $1$ and $1$ .

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

Step 5

Simplify terms.

Tap for more steps...

Step 5.1

Rewrite using the commutative property of multiplication.

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

Step 5.2

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

Tap for more steps...

Step 5.2.1

Move the leading negative in $- \frac{1}{\sin (x)}$ into the numerator.

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

Step 5.2.2

Cancel the common factor.

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

Step 5.2.3

Rewrite the expression.

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

Step 5.3

Simplify each term.

Tap for more steps...

Step 5.3.1

Rewrite $1$ as $1^{2}$ .

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

Step 5.3.2

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

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

Step 5.3.3

Convert from $\frac{1}{\sin (x)}$ to $\csc (x)$ .

$\csc^{2} (x) - 1$

Step 6

Apply pythagorean identity.

$\cot^{2} (x)$