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Simplifying Complex Fractions

Complex fractions are fractions in which the numerator or denominator, or both, contain fractions. They are critical to
analytic trigonometry and identity proving, because each individual trigonometric function can be expressed as a
fraction involving other trigonometric functions . Whenever a complex fraction appears, you should simplify it
immediately, before you use it in any other operation .
Otherwise, you risk making a straightforward problem
exponentially more difficult .

There are 2 primary ways to simplify complex fractions.

METHOD 1: Write the numerator and denominator each as a single fraction, then perform the implied division .

 
Find the LCD for the numerator (c), and find the LCD for the denominator (cs)
Perform the addition /subtraction in the numerator and denominator
Multiply by the reciprocal of the denominator
Cancel and collect remaining factors

METHOD 2: Multiply the numerator and denominator by the LCD of all the component fractions.

 
Find the LCD of (from numerator), and (from denominator)
Multiply the LCD into the numerator and denominator, using distribution as needed
Factor the numerator and denominator, and cancel if needed

You should try both, and decide which is more natural for you.

Simplify the following.
NOTE: c and s represent variables . Do NOT assume they mean cos x or sin x .

Some possible answers to choose from:
NOTE: half of these answers are not correct of any of the problems above

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