Trigonometry Examples

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

Step 1

Simplify each term.

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Step 1.1

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

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

Step 1.2

Move $- 2$ .

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

Step 1.3

Reorder $2 \cos^{2} (x)$ and $- 2$ .

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

Step 1.4

Factor $- 2$ out of $- 2$ .

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

Step 1.5

Factor $- 2$ out of $2 \cos^{2} (x)$ .

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

Step 1.6

Factor $- 2$ out of $- 2 (1) - 2 (- \cos^{2} (x))$ .

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

Step 1.7

Apply pythagorean identity.

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

Step 1.8

Simplify each term.

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Step 1.8.1

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

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

Step 1.8.2

Apply the product rule to $\frac{1}{\cos (x)}$ .

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

Step 1.8.3

One to any power is one.

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

Step 1.8.4

Combine $2$ and $\frac{1}{\cos^{2} (x)}$ .

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

Step 1.9

Apply the distributive property.

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

Step 1.10

Rewrite using the commutative property of multiplication.

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

Step 1.11

Combine.

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

Step 1.12

Multiply $1$ by $2$ .

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

Step 1.13

Simplify each term.

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Step 1.13.1

Combine $- 2$ and $\frac{1}{\cos (x)}$ .

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

Step 1.13.2

Move the negative in front of the fraction.

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

Step 1.13.3

Combine $\sin^{2} (x)$ and $\frac{2}{\cos (x)}$ .

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

Step 1.13.4

Move $2$ to the left of $\sin^{2} (x)$ .

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

Step 1.13.5

Multiply $\cos (x)$ by $\cos^{2} (x)$ by adding the exponents.

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Step 1.13.5.1

Multiply $\cos (x)$ by $\cos^{2} (x)$ .

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Step 1.13.5.1.1

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

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

Step 1.13.5.1.2

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

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

Step 1.13.5.2

Add $1$ and $2$ .

$\cos (x) - \frac{2 \sin^{2} (x)}{\cos (x)} + \frac{2}{\cos^{3} (x)}$

Step 2

Simplify each term.

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Step 2.1

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

$\cos (x) - \frac{2 (\sin (x) \sin (x))}{\cos (x)} + \frac{2}{\cos^{3} (x)}$

Step 2.2

Separate fractions.

$\cos (x) - (\frac{2 (\sin (x))}{1} \cdot \frac{\sin (x)}{\cos (x)}) + \frac{2}{\cos^{3} (x)}$

Step 2.3

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

$\cos (x) - (\frac{2 (\sin (x))}{1} \tan (x)) + \frac{2}{\cos^{3} (x)}$

Step 2.4

Divide $2 (\sin (x))$ by $1$ .

$\cos (x) - (2 (\sin (x)) \tan (x)) + \frac{2}{\cos^{3} (x)}$

Step 2.5

Multiply $2$ by $- 1$ .

$\cos (x) - 2 (\sin (x) \tan (x)) + \frac{2}{\cos^{3} (x)}$

Step 2.6

Multiply by $1$ .

$\cos (x) - 2 \sin (x) \tan (x) + \frac{2}{\cos^{3} (x) \cdot 1}$

Step 2.7

Separate fractions.

$\cos (x) - 2 \sin (x) \tan (x) + \frac{2}{1} \cdot \frac{1}{\cos^{3} (x)}$

Step 2.8

Convert from $\frac{1}{\cos^{3} (x)}$ to $\sec^{3} (x)$ .

$\cos (x) - 2 \sin (x) \tan (x) + \frac{2}{1} \sec^{3} (x)$

Step 2.9

Divide $2$ by $1$ .

$\cos (x) - 2 \sin (x) \tan (x) + 2 \sec^{3} (x)$