Trigonometry Examples

$\frac{\sin (x) - \cos (x)}{\cos (x)} + \frac{\sin (x) + \cos (x)}{\sin (x)} = \csc (x) \sec (x)$

Step 1

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

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Step 1.1

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

$\frac{\sin (x) - \cos (x)}{\cos (x)} \cdot \frac{\sin (x)}{\sin (x)} + \frac{\sin (x) + \cos (x)}{\sin (x)} = \csc (x) \sec (x)$

Step 1.2

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

$\frac{\sin (x) - \cos (x)}{\cos (x)} \cdot \frac{\sin (x)}{\sin (x)} + \frac{\sin (x) + \cos (x)}{\sin (x)} \cdot \frac{\cos (x)}{\cos (x)} = \csc (x) \sec (x)$

Step 1.3

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

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Step 1.3.1

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

$\frac{(\sin (x) - \cos (x)) \sin (x)}{\cos (x) \sin (x)} + \frac{\sin (x) + \cos (x)}{\sin (x)} \cdot \frac{\cos (x)}{\cos (x)} = \csc (x) \sec (x)$

Step 1.3.2

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

$\frac{(\sin (x) - \cos (x)) \sin (x)}{\cos (x) \sin (x)} + \frac{(\sin (x) + \cos (x)) \cos (x)}{\sin (x) \cos (x)} = \csc (x) \sec (x)$

Step 1.3.3

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

$\frac{(\sin (x) - \cos (x)) \sin (x)}{\cos (x) \sin (x)} + \frac{(\sin (x) + \cos (x)) \cos (x)}{\cos (x) \sin (x)} = \csc (x) \sec (x)$

Step 1.4

Combine the numerators over the common denominator.

$\frac{(\sin (x) - \cos (x)) \sin (x) + (\sin (x) + \cos (x)) \cos (x)}{\cos (x) \sin (x)} = \csc (x) \sec (x)$

Step 1.5

Simplify the numerator.

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Step 1.5.1

Apply the distributive property.

$\frac{\sin (x) \sin (x) - \cos (x) \sin (x) + (\sin (x) + \cos (x)) \cos (x)}{\cos (x) \sin (x)} = \csc (x) \sec (x)$

Step 1.5.2

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

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Step 1.5.2.1

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

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

Step 1.5.2.2

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

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

Step 1.5.2.3

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

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

Step 1.5.2.4

Add $1$ and $1$ .

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

Step 1.5.3

Apply the distributive property.

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

Step 1.5.4

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

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Step 1.5.4.1

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

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

Step 1.5.4.2

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

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

Step 1.5.4.3

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

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

Step 1.5.4.4

Add $1$ and $1$ .

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

Step 1.5.5

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

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

Step 1.5.6

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

$\frac{\sin^{2} (x) + 0 + \cos^{2} (x)}{\cos (x) \sin (x)} = \csc (x) \sec (x)$

Step 1.5.7

Add $\sin^{2} (x)$ and $0$ .

$\frac{\sin^{2} (x) + \cos^{2} (x)}{\cos (x) \sin (x)} = \csc (x) \sec (x)$

Step 1.5.8

Apply pythagorean identity.

$\frac{1}{\cos (x) \sin (x)} = \csc (x) \sec (x)$

Step 1.6

Separate fractions.

$\frac{1}{\sin (x)} \cdot \frac{1}{\cos (x)} = \csc (x) \sec (x)$

Step 1.7

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

$\frac{1}{\sin (x)} \sec (x) = \csc (x) \sec (x)$

Step 1.8

Convert from $\frac{1}{\sin (x)}$ to $\csc (x)$ .

$\csc (x) \sec (x) = \csc (x) \sec (x)$

Step 2

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

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

Step 3