Precalculus Examples

$\csc (A) - \cos (A) \cot (A)$

Step 1

Simplify terms.

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Step 1.1

Simplify each term.

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Step 1.1.1

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

$\frac{1}{\sin (A)} - \cos (A) \cot (A)$

Step 1.1.2

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

$\frac{1}{\sin (A)} - \cos (A) \frac{\cos (A)}{\sin (A)}$

Step 1.1.3

Multiply $- \cos (A) \frac{\cos (A)}{\sin (A)}$ .

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Step 1.1.3.1

Combine $\frac{\cos (A)}{\sin (A)}$ and $\cos (A)$ .

$\frac{1}{\sin (A)} - \frac{\cos (A) \cos (A)}{\sin (A)}$

Step 1.1.3.2

Raise $\cos (A)$ to the power of $1$ .

$\frac{1}{\sin (A)} - \frac{\cos^{1} (A) \cos (A)}{\sin (A)}$

Step 1.1.3.3

Raise $\cos (A)$ to the power of $1$ .

$\frac{1}{\sin (A)} - \frac{\cos^{1} (A) \cos^{1} (A)}{\sin (A)}$

Step 1.1.3.4

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

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

Step 1.1.3.5

Add $1$ and $1$ .

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

Step 1.2

Combine the numerators over the common denominator.

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

Step 2

Apply pythagorean identity.

$\frac{\sin^{2} (A)}{\sin (A)}$

Step 3

Cancel the common factor of $\sin^{2} (A)$ and $\sin (A)$ .

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Step 3.1

Factor $\sin (A)$ out of $\sin^{2} (A)$ .

$\frac{\sin (A) \sin (A)}{\sin (A)}$

Step 3.2

Cancel the common factors.

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Step 3.2.1

Multiply by $1$ .

$\frac{\sin (A) \sin (A)}{\sin (A) \cdot 1}$

Step 3.2.2

Cancel the common factor.

$\frac{\sin (A) \sin (A)}{\sin (A) \cdot 1}$

Step 3.2.3

Rewrite the expression.

$\frac{\sin (A)}{1}$

Step 3.2.4

Divide $\sin (A)$ by $1$ .

$\sin (A)$