Trigonometry Examples

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

Step 1

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

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

Step 2

To write $\frac{\cos (x)}{\sin (x)}$ as a fraction with a common denominator, multiply by $\frac{\cos (x)}{\cos (x)}$ .

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

Step 3

Simplify terms.

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Step 3.1

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

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

Step 3.2

Combine the numerators over the common denominator.

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

Step 4

Simplify the numerator.

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Step 4.1

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

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Step 4.1.1

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

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

Step 4.1.2

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

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

Step 4.1.3

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

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

Step 4.1.4

Add $1$ and $1$ .

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

Step 4.2

Subtract $2 \cos^{2} (x)$ from $\cos^{2} (x)$ .

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

Step 4.3

Rewrite $1 - \cos^{2} (x)$ in a factored form.

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Step 4.3.1

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

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

Step 4.3.2

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = 1$ and $b = \cos (x)$ .

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