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

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

Step 1

Rewrite ${(\csc (x) - \sec (x))}^{2}$ as $(\csc (x) - \sec (x)) (\csc (x) - \sec (x))$ .

$(\csc (x) - \sec (x)) (\csc (x) - \sec (x))$

Step 2

Expand $(\csc (x) - \sec (x)) (\csc (x) - \sec (x))$ using the FOIL Method.

Apply the distributive property.

$\csc (x) (\csc (x) - \sec (x)) - \sec (x) (\csc (x) - \sec (x))$

Apply the distributive property.

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

Apply the distributive property.

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

Step 3

Simplify and combine like terms.

Simplify each term.

Multiply $\csc (x) \csc (x)$ .

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

$\csc^{1} (x) \csc (x) + \csc (x) (- \sec (x)) - \sec (x) \csc (x) - \sec (x) (- \sec (x))$

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

$\csc^{1} (x) \csc^{1} (x) + \csc (x) (- \sec (x)) - \sec (x) \csc (x) - \sec (x) (- \sec (x))$

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

${\csc (x)}^{1 + 1} + \csc (x) (- \sec (x)) - \sec (x) \csc (x) - \sec (x) (- \sec (x))$

Add $1$ and $1$ .

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

Rewrite using the commutative property of multiplication.

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

Multiply $- \sec (x) (- \sec (x))$ .

Multiply $- 1$ by $- 1$ .

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

Multiply $\sec (x)$ by $1$ .

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

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

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

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

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

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

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

Add $1$ and $1$ .

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

Reorder the factors of $- \sec (x) \csc (x)$ .

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

Subtract $\csc (x) \sec (x)$ from $- \csc (x) \sec (x)$ .

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