Trigonometry Examples

$\cos^{2} (x) - \cot^{2} (x) = - \cos^{2} (x) \cot^{2} (x)$

Step 1

Simplify $\cos^{2} (x) - \cot^{2} (x)$ .

Tap for more steps...

Step 1.1

Simplify terms.

Tap for more steps...

Step 1.1.1

Simplify each term.

Tap for more steps...

Step 1.1.1.1

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

$\cos^{2} (x) - {(\frac{\cos (x)}{\sin (x)})}^{2} = - \cos^{2} (x) \cot^{2} (x)$

Step 1.1.1.2

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

$\cos^{2} (x) - \frac{\cos^{2} (x)}{\sin^{2} (x)} = - \cos^{2} (x) \cot^{2} (x)$

Step 1.1.2

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

$\cos^{2} (x) - {(\frac{\cos (x)}{\sin (x)})}^{2} = - \cos^{2} (x) \cot^{2} (x)$

Step 1.2

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

$(\cos (x) + \frac{\cos (x)}{\sin (x)}) (\cos (x) - \frac{\cos (x)}{\sin (x)}) = - \cos^{2} (x) \cot^{2} (x)$

Step 1.3

Convert from $\frac{\cos (x)}{\sin (x)}$ to $\cot (x)$ .

$(\cos (x) + \cot (x)) (\cos (x) - \frac{\cos (x)}{\sin (x)}) = - \cos^{2} (x) \cot^{2} (x)$

Step 1.4

Convert from $\frac{\cos (x)}{\sin (x)}$ to $\cot (x)$ .

$(\cos (x) + \cot (x)) (\cos (x) - \cot (x)) = - \cos^{2} (x) \cot^{2} (x)$

Step 1.5

Expand $(\cos (x) + \cot (x)) (\cos (x) - \cot (x))$ using the FOIL Method.

Tap for more steps...

Step 1.5.1

Apply the distributive property.

$\cos (x) (\cos (x) - \cot (x)) + \cot (x) (\cos (x) - \cot (x)) = - \cos^{2} (x) \cot^{2} (x)$

Step 1.5.2

Apply the distributive property.

$\cos (x) \cos (x) + \cos (x) (- \cot (x)) + \cot (x) (\cos (x) - \cot (x)) = - \cos^{2} (x) \cot^{2} (x)$

Step 1.5.3

Apply the distributive property.

$\cos (x) \cos (x) + \cos (x) (- \cot (x)) + \cot (x) \cos (x) + \cot (x) (- \cot (x)) = - \cos^{2} (x) \cot^{2} (x)$

Step 1.6

Simplify terms.

Tap for more steps...

Step 1.6.1

Combine the opposite terms in $\cos (x) \cos (x) + \cos (x) (- \cot (x)) + \cot (x) \cos (x) + \cot (x) (- \cot (x))$ .

Tap for more steps...

Step 1.6.1.1

Reorder the factors in the terms $\cos (x) (- \cot (x))$ and $\cot (x) \cos (x)$ .

$\cos (x) \cos (x) - \cos (x) \cot (x) + \cos (x) \cot (x) + \cot (x) (- \cot (x)) = - \cos^{2} (x) \cot^{2} (x)$

Step 1.6.1.2

Add $- \cos (x) \cot (x)$ and $\cos (x) \cot (x)$ .

$\cos (x) \cos (x) + 0 + \cot (x) (- \cot (x)) = - \cos^{2} (x) \cot^{2} (x)$

Step 1.6.1.3

Add $\cos (x) \cos (x)$ and $0$ .

$\cos (x) \cos (x) + \cot (x) (- \cot (x)) = - \cos^{2} (x) \cot^{2} (x)$

Step 1.6.2

Simplify each term.

Tap for more steps...

Step 1.6.2.1

Multiply $\cos (x) \cos (x)$ .

Tap for more steps...

Step 1.6.2.1.1

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

$\cos^{1} (x) \cos (x) + \cot (x) (- \cot (x)) = - \cos^{2} (x) \cot^{2} (x)$

Step 1.6.2.1.2

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

$\cos^{1} (x) \cos^{1} (x) + \cot (x) (- \cot (x)) = - \cos^{2} (x) \cot^{2} (x)$

Step 1.6.2.1.3

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

${\cos (x)}^{1 + 1} + \cot (x) (- \cot (x)) = - \cos^{2} (x) \cot^{2} (x)$

Step 1.6.2.1.4

Add $1$ and $1$ .

$\cos^{2} (x) + \cot (x) (- \cot (x)) = - \cos^{2} (x) \cot^{2} (x)$

Step 1.6.2.2

Rewrite using the commutative property of multiplication.

$\cos^{2} (x) - \cot (x) \cot (x) = - \cos^{2} (x) \cot^{2} (x)$

Step 1.6.2.3

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

Tap for more steps...

Step 1.6.2.3.1

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

$\cos^{2} (x) - (\cot^{1} (x) \cot (x)) = - \cos^{2} (x) \cot^{2} (x)$

Step 1.6.2.3.2

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

$\cos^{2} (x) - (\cot^{1} (x) \cot^{1} (x)) = - \cos^{2} (x) \cot^{2} (x)$

Step 1.6.2.3.3

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

$\cos^{2} (x) - {\cot (x)}^{1 + 1} = - \cos^{2} (x) \cot^{2} (x)$

Step 1.6.2.3.4

Add $1$ and $1$ .

$\cos^{2} (x) - \cot^{2} (x) = - \cos^{2} (x) \cot^{2} (x)$

Step 2

Graph each side of the equation. The solution is the x-value of the point of intersection.

$x \approx - 0.1, 0.1, \frac{1}{5}, - 0.3, 0.3, \frac{2}{5}, - \frac{1}{2}$

Step 3

The result can be shown in multiple forms.

Exact Form:

$x \approx - 0.1, 0.1, \frac{1}{5}, - 0.3, 0.3, \frac{2}{5}, - \frac{1}{2}$

Decimal Form:

$x \approx - 0.1, 0.1, 0.2, - 0.3, 0.3, 0.4, - 0.5$

Step 4