Trigonometry Examples

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

Step 1

Start on the left side.

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

Step 2

Simplify the expression.

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Step 2.1

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

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

Step 2.2

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

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

Step 2.3

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

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

Step 2.4

Write each expression with a common denominator of $\cos (x) (1 + \sin (x))$ , by multiplying each by an appropriate factor of $1$ .

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Step 2.4.1

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

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

Step 2.4.2

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

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

Step 2.4.3

Reorder the factors of $(1 + \sin (x)) \cos (x)$ .

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

Step 2.5

Combine the numerators over the common denominator.

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

Step 2.6

Simplify the numerator.

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Step 2.6.1

Apply the distributive property.

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

Step 2.6.2

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

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

Step 2.6.3

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

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Step 2.6.3.1

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

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

Step 2.6.3.2

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

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

Step 2.6.3.3

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

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

Step 2.6.3.4

Add $1$ and $1$ .

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

Step 2.6.4

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

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Step 2.6.4.1

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

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

Step 2.6.4.2

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

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

Step 2.6.4.3

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

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

Step 2.6.4.4

Add $1$ and $1$ .

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

Step 2.6.5

Apply pythagorean identity.

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

Step 2.7

Cancel the common factor of $\sin (x) + 1$ and $1 + \sin (x)$ .

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Step 2.7.1

Reorder terms.

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

Step 2.7.2

Cancel the common factor.

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

Step 2.7.3

Rewrite the expression.

$\frac{1}{\cos (x)}$

Step 3

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

$\tan (x) + \frac{\cos (x)}{1 + \sin (x)} = \frac{1}{\cos (x)}$ is an identity