Trigonometry Examples

$\cos (x) (\tan (x) + 2) (2 \tan (x) + 1) = 2 \sec (x) + 5 \sin (x)$

Step 1

Start on the left side.

$\cos (x) (\tan (x) + 2) (2 \tan (x) + 1)$

Step 2

Simplify the expression.

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Step 2.1

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

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

Step 2.2

Apply the distributive property.

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

Step 2.3

Cancel the common factor of $\cos (x)$ .

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Step 2.3.1

Cancel the common factor.

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

Step 2.3.2

Rewrite the expression.

$(\sin (x) + \cos (x) \cdot 2) (2 \tan (x) + 1)$

Step 2.4

Move $2$ to the left of $\cos (x)$ .

$(\sin (x) + 2 \cdot \cos (x)) (2 \tan (x) + 1)$

Step 2.5

Simplify each term.

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Step 2.5.1

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

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

Step 2.5.2

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

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

Step 2.6

Expand $(\sin (x) + 2 \cos (x)) (\frac{2 \sin (x)}{\cos (x)} + 1)$ using the FOIL Method.

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Step 2.6.1

Apply the distributive property.

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

Step 2.6.2

Apply the distributive property.

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

Step 2.6.3

Apply the distributive property.

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

Step 2.7

Simplify and combine like terms.

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Step 2.7.1

Simplify each term.

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Step 2.7.1.1

Multiply $\sin (x) \frac{2 \sin (x)}{\cos (x)}$ .

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Step 2.7.1.1.1

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

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

Step 2.7.1.1.2

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

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

Step 2.7.1.1.3

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

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

Step 2.7.1.1.4

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

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

Step 2.7.1.1.5

Add $1$ and $1$ .

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

Step 2.7.1.2

Multiply $\sin (x)$ by $1$ .

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

Step 2.7.1.3

Cancel the common factor of $\cos (x)$ .

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Step 2.7.1.3.1

Factor $\cos (x)$ out of $2 \cos (x)$ .

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

Step 2.7.1.3.2

Cancel the common factor.

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

Step 2.7.1.3.3

Rewrite the expression.

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

Step 2.7.1.4

Multiply $2$ by $2$ .

$\frac{2 \sin^{2} (x)}{\cos (x)} + \sin (x) + 4 \sin (x) + 2 \cos (x) \cdot 1$

Step 2.7.1.5

Multiply $2$ by $1$ .

$\frac{2 \sin^{2} (x)}{\cos (x)} + \sin (x) + 4 \sin (x) + 2 \cos (x)$

Step 2.7.2

Add $\sin (x)$ and $4 \sin (x)$ .

$\frac{2 \sin^{2} (x)}{\cos (x)} + 5 \sin (x) + 2 \cos (x)$

Step 3

Apply Pythagorean identity in reverse.

$\frac{2 (1 - \cos^{2} (x))}{\cos (x)} + 5 \sin (x) + 2 \cos (x)$

Step 4

Simplify.

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Step 4.1

Simplify the numerator.

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Step 4.1.1

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

$\frac{2 (1^{2} - {\cos (x)}^{2})}{\cos (x)} + 5 \sin (x) + 2 \cos (x)$

Step 4.1.2

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

$\frac{2 (1 + \cos (x)) (1 - \cos (x))}{\cos (x)} + 5 \sin (x) + 2 \cos (x)$

Step 4.2

To write $5 \sin (x)$ as a fraction with a common denominator, multiply by $\frac{\cos (x)}{\cos (x)}$ .

$\frac{2 (1 + \cos (x)) (1 - \cos (x))}{\cos (x)} + \frac{5 \sin (x) \cos (x)}{\cos (x)} + 2 \cos (x)$

Step 4.3

Combine the numerators over the common denominator.

$\frac{2 (1 + \cos (x)) (1 - \cos (x)) + 5 \sin (x) \cos (x)}{\cos (x)} + 2 \cos (x)$

Step 4.4

Simplify the numerator.

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Step 4.4.1

Apply the distributive property.

$\frac{(2 \cdot 1 + 2 \cos (x)) (1 - \cos (x)) + 5 \sin (x) \cos (x)}{\cos (x)} + 2 \cos (x)$

Step 4.4.2

Multiply $2$ by $1$ .

$\frac{(2 + 2 \cos (x)) (1 - \cos (x)) + 5 \sin (x) \cos (x)}{\cos (x)} + 2 \cos (x)$

Step 4.4.3

Expand $(2 + 2 \cos (x)) (1 - \cos (x))$ using the FOIL Method.

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Step 4.4.3.1

Apply the distributive property.

$\frac{2 (1 - \cos (x)) + 2 \cos (x) (1 - \cos (x)) + 5 \sin (x) \cos (x)}{\cos (x)} + 2 \cos (x)$

Step 4.4.3.2

Apply the distributive property.

$\frac{2 \cdot 1 + 2 (- \cos (x)) + 2 \cos (x) (1 - \cos (x)) + 5 \sin (x) \cos (x)}{\cos (x)} + 2 \cos (x)$

Step 4.4.3.3

Apply the distributive property.

$\frac{2 \cdot 1 + 2 (- \cos (x)) + 2 \cos (x) \cdot 1 + 2 \cos (x) (- \cos (x)) + 5 \sin (x) \cos (x)}{\cos (x)} + 2 \cos (x)$

Step 4.4.4

Simplify and combine like terms.

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Step 4.4.4.1

Simplify each term.

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Step 4.4.4.1.1

Multiply $2$ by $1$ .

$\frac{2 + 2 (- \cos (x)) + 2 \cos (x) \cdot 1 + 2 \cos (x) (- \cos (x)) + 5 \sin (x) \cos (x)}{\cos (x)} + 2 \cos (x)$

Step 4.4.4.1.2

Multiply $- 1$ by $2$ .

$\frac{2 - 2 \cos (x) + 2 \cos (x) \cdot 1 + 2 \cos (x) (- \cos (x)) + 5 \sin (x) \cos (x)}{\cos (x)} + 2 \cos (x)$

Step 4.4.4.1.3

Multiply $2$ by $1$ .

$\frac{2 - 2 \cos (x) + 2 \cos (x) + 2 \cos (x) (- \cos (x)) + 5 \sin (x) \cos (x)}{\cos (x)} + 2 \cos (x)$

Step 4.4.4.1.4

Multiply $2 \cos (x) (- \cos (x))$ .

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Step 4.4.4.1.4.1

Multiply $- 1$ by $2$ .

$\frac{2 - 2 \cos (x) + 2 \cos (x) - 2 \cos (x) \cos (x) + 5 \sin (x) \cos (x)}{\cos (x)} + 2 \cos (x)$

Step 4.4.4.1.4.2

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

$\frac{2 - 2 \cos (x) + 2 \cos (x) - 2 ({\cos (x)}^{1} \cos (x)) + 5 \sin (x) \cos (x)}{\cos (x)} + 2 \cos (x)$

Step 4.4.4.1.4.3

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

$\frac{2 - 2 \cos (x) + 2 \cos (x) - 2 ({\cos (x)}^{1} {\cos (x)}^{1}) + 5 \sin (x) \cos (x)}{\cos (x)} + 2 \cos (x)$

Step 4.4.4.1.4.4

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

$\frac{2 - 2 \cos (x) + 2 \cos (x) - 2 {\cos (x)}^{1 + 1} + 5 \sin (x) \cos (x)}{\cos (x)} + 2 \cos (x)$

Step 4.4.4.1.4.5

Add $1$ and $1$ .

$\frac{2 - 2 \cos (x) + 2 \cos (x) - 2 {\cos (x)}^{2} + 5 \sin (x) \cos (x)}{\cos (x)} + 2 \cos (x)$

Step 4.4.4.2

Add $- 2 \cos (x)$ and $2 \cos (x)$ .

$\frac{2 + 0 - 2 {\cos (x)}^{2} + 5 \sin (x) \cos (x)}{\cos (x)} + 2 \cos (x)$

Step 4.4.4.3

Add $2$ and $0$ .

$\frac{2 - 2 {\cos (x)}^{2} + 5 \sin (x) \cos (x)}{\cos (x)} + 2 \cos (x)$

Step 4.5

To write $2 \cos (x)$ as a fraction with a common denominator, multiply by $\frac{\cos (x)}{\cos (x)}$ .

$\frac{2 - 2 {\cos (x)}^{2} + 5 \sin (x) \cos (x)}{\cos (x)} + \frac{2 \cos (x) \cos (x)}{\cos (x)}$

Step 4.6

Combine the numerators over the common denominator.

$\frac{2 - 2 {\cos (x)}^{2} + 5 \sin (x) \cos (x) + 2 \cos (x) \cos (x)}{\cos (x)}$

Step 4.7

Simplify the numerator.

$\frac{2 + 5 \sin (x) \cos (x)}{\cos (x)}$

Step 5

Now consider the right side of the equation.

$2 \sec (x) + 5 \sin (x)$

Step 6

Apply the reciprocal identity to $\sec (x)$ .

$2 \frac{1}{\cos (x)} + 5 \sin (x)$

Step 7

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

$\frac{2}{\cos (x)} + 5 \sin (x)$

Step 8

Write $5 \sin (x)$ as a fraction with denominator $1$ .

$\frac{2}{\cos (x)} + \frac{5 \sin (x)}{1}$

Step 9

Add fractions.

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Step 9.1

To write $\frac{5 \sin (x)}{1}$ as a fraction with a common denominator, multiply by $\frac{\cos (x)}{\cos (x)}$ .

$\frac{2}{\cos (x)} + \frac{5 \sin (x)}{1} \cdot \frac{\cos (x)}{\cos (x)}$

Step 9.2

Multiply $\frac{5 \sin (x)}{1}$ by $\frac{\cos (x)}{\cos (x)}$ .

$\frac{2}{\cos (x)} + \frac{5 \sin (x) \cos (x)}{\cos (x)}$

Step 9.3

Combine the numerators over the common denominator.

$\frac{2 + 5 \sin (x) \cos (x)}{\cos (x)}$

Step 10

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

$\cos (x) (\tan (x) + 2) (2 \tan (x) + 1) = 2 \sec (x) + 5 \sin (x)$ is an identity