Simplify (sin(x)-csc(x))^2

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

Step 1

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

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

Step 2

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

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

Step 3

Expand $(\sin (x) - \frac{1}{\sin (x)}) (\sin (x) - \frac{1}{\sin (x)})$ using the FOIL Method.

Apply the distributive property.

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

Apply the distributive property.

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

Apply the distributive property.

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

Step 4

Simplify and combine like terms.

Simplify each term.

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

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

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

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

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

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

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

Add $1$ and $1$ .

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

Rewrite using the commutative property of multiplication.

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

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

Factor $\sin (x)$ out of $- \sin (x)$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

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

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

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

Cancel the common factor.

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

Rewrite the expression.

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

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

Multiply $- 1$ by $- 1$ .

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

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

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

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

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

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

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

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

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

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

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

Add $1$ and $1$ .

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

Subtract $1$ from $- 1$ .

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

Step 5

Simplify each term.

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

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

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

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

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

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

Simplify (sin(x)-csc(x))^2

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