Precalculus Examples

$\csc (θ) (\csc (θ) + \sin (- θ))$

Step 1

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

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

Step 2

Simplify each term.

Tap for more steps...

Step 2.1

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

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

Step 2.2

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

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

Step 3

Apply the distributive property.

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

Step 4

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

Tap for more steps...

Step 4.1

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

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

Step 4.2

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

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

Step 4.3

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

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

Step 4.4

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

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

Step 4.5

Add $1$ and $1$ .

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

Step 5

Simplify terms.

Tap for more steps...

Step 5.1

Rewrite using the commutative property of multiplication.

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

Step 5.2

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

Tap for more steps...

Step 5.2.1

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

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

Step 5.2.2

Cancel the common factor.

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

Step 5.2.3

Rewrite the expression.

$\frac{1}{\sin^{2} (θ)} - 1$

Step 5.3

Simplify each term.

Tap for more steps...

Step 5.3.1

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

$\frac{1^{2}}{\sin^{2} (θ)} - 1$

Step 5.3.2

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

${(\frac{1}{\sin (θ)})}^{2} - 1$

Step 5.3.3

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

$\csc^{2} (θ) - 1$

Step 6

Apply pythagorean identity.

$\cot^{2} (θ)$