Precalculus Examples

$\cot^{2} (A) + \csc^{2} (A) = - \cot^{4} (A) + \csc^{4} (A)$

Step 1

Start on the right side.

$- \cot^{4} (A) + \csc^{4} (A)$

Step 2

Simplify the expression.

Tap for more steps...

Step 2.1

Rewrite $\csc^{4} (A)$ as ${(\csc^{2} (A))}^{2}$ .

$- \cot^{4} (A) + {(\csc^{2} (A))}^{2}$

Step 2.2

Rewrite $\cot^{4} (A)$ as ${(\cot^{2} (A))}^{2}$ .

$- {(\cot^{2} (A))}^{2} + {(\csc^{2} (A))}^{2}$

Step 2.3

Reorder $- {(\cot^{2} (A))}^{2}$ and ${(\csc^{2} (A))}^{2}$ .

${(\csc^{2} (A))}^{2} - {(\cot^{2} (A))}^{2}$

Step 2.4

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = \csc^{2} (A)$ and $b = \cot^{2} (A)$ .

$(\csc^{2} (A) + \cot^{2} (A)) (\csc^{2} (A) - \cot^{2} (A))$

Step 2.5

Simplify each term.

Tap for more steps...

Step 2.5.1

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

$({(\frac{1}{\sin (A)})}^{2} + \cot^{2} (A)) (\csc^{2} (A) - \cot^{2} (A))$

Step 2.5.2

Apply the product rule to $\frac{1}{\sin (A)}$ .

$(\frac{1^{2}}{\sin^{2} (A)} + \cot^{2} (A)) (\csc^{2} (A) - \cot^{2} (A))$

Step 2.5.3

One to any power is one.

$(\frac{1}{\sin^{2} (A)} + \cot^{2} (A)) (\csc^{2} (A) - \cot^{2} (A))$

Step 2.5.4

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

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

Step 2.5.5

Apply the product rule to $\frac{\cos (A)}{\sin (A)}$ .

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

Step 2.6

Apply pythagorean identity.

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

Step 2.7

Multiply $\frac{1}{\sin^{2} (A)} + \frac{\cos^{2} (A)}{\sin^{2} (A)}$ by $1$ .

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

Step 3

Combine the numerators over the common denominator.

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

Step 4

Now consider the left side of the equation.

$\cot^{2} (A) + \csc^{2} (A)$

Step 5

Convert to sines and cosines.

Tap for more steps...

Step 5.1

Write $\cot (A)$ in sines and cosines using the quotient identity.

${(\frac{\cos (A)}{\sin (A)})}^{2} + \csc^{2} (A)$

Step 5.2

Apply the reciprocal identity to $\csc (A)$ .

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

Step 5.3

Apply the product rule to $\frac{\cos (A)}{\sin (A)}$ .

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

Step 5.4

Apply the product rule to $\frac{1}{\sin (A)}$ .

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

Step 6

One to any power is one.

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

Step 7

Combine the numerators over the common denominator.

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

Step 8

Reorder terms.

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

Step 9

Because the two sides have been shown to be equivalent, the equation is an identity.

$\cot^{2} (A) + \csc^{2} (A) = - \cot^{4} (A) + \csc^{4} (A)$ is an identity