Trigonometry Examples

\frac{\cos (a - b)}{\cos (a) \cos (b)}

$\frac{\cos (a - b)}{\cos (a) \cos (b)}$

Step 1

Separate fractions.

\frac{1}{\cos (b)} \cdot \frac{\cos (a - b)}{\cos (a)}

$\frac{1}{\cos (b)} \cdot \frac{\cos (a - b)}{\cos (a)}$

Step 2

Rewrite

\frac{\cos (a - b)}{\cos (a)}

$\frac{\cos (a - b)}{\cos (a)}$ as a product.

\frac{1}{\cos (b)} (\cos (a - b) \frac{1}{\cos (a)})

$\frac{1}{\cos (b)} (\cos (a - b) \frac{1}{\cos (a)})$

Step 3

Write

\cos (a - b)

$\cos (a - b)$ as a fraction with denominator

1

$1$ .

\frac{1}{\cos (b)} (\frac{\cos (a - b)}{1} \cdot \frac{1}{\cos (a)})

$\frac{1}{\cos (b)} (\frac{\cos (a - b)}{1} \cdot \frac{1}{\cos (a)})$

Step 4

Simplify.

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Step 4.1

Divide

\cos (a - b)

$\cos (a - b)$ by

1

$1$ .

\frac{1}{\cos (b)} (\cos (a - b) \frac{1}{\cos (a)})

$\frac{1}{\cos (b)} (\cos (a - b) \frac{1}{\cos (a)})$

Step 4.2

Convert from

\frac{1}{\cos (a)}

$\frac{1}{\cos (a)}$ to

\sec (a)

$\sec (a)$ .

\frac{1}{\cos (b)} (\cos (a - b) \sec (a))

$\frac{1}{\cos (b)} (\cos (a - b) \sec (a))$

\frac{1}{\cos (b)} (\cos (a - b) \sec (a))

$\frac{1}{\cos (b)} (\cos (a - b) \sec (a))$

Step 5

Convert from

\frac{1}{\cos (b)}

$\frac{1}{\cos (b)}$ to

\sec (b)

$\sec (b)$ .

\sec (b) (\cos (a - b) \sec (a))

$\sec (b) (\cos (a - b) \sec (a))$

Step 6

Apply the difference of angles identity

\cos (x - y) = \cos (x) \cos (y) + \sin (x) \sin (y)

$\cos (x - y) = \cos (x) \cos (y) + \sin (x) \sin (y)$ .

\sec (b) ((\cos (a) \cos (b) + \sin (a) \sin (b)) \sec (a))

$\sec (b) ((\cos (a) \cos (b) + \sin (a) \sin (b)) \sec (a))$

Step 7

Simplify by multiplying through.

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Step 7.1

Apply the distributive property.

\sec (b) (\cos (a) \cos (b) \sec (a) + \sin (a) \sin (b) \sec (a))

$\sec (b) (\cos (a) \cos (b) \sec (a) + \sin (a) \sin (b) \sec (a))$

Step 7.2

Apply the distributive property.

\sec (b) \cos (a) \cos (b) \sec (a) + \sec (b) \sin (a) \sin (b) \sec (a)

$\sec (b) \cos (a) \cos (b) \sec (a) + \sec (b) \sin (a) \sin (b) \sec (a)$

\sec (b) \cos (a) \cos (b) \sec (a) + \sec (b) \sin (a) \sin (b) \sec (a)

$\sec (b) \cos (a) \cos (b) \sec (a) + \sec (b) \sin (a) \sin (b) \sec (a)$