Trig Identities - Day 2

And now it's time to learn how to verify trig identities! - a.k.a. TRIG PROOFS! (insert ominous music here)

To do this you are given two things that are already equal to each other but are still forced to prove that they are equal through the use of tiny baby steps. The end goal of any trig identity verification is to make the left side of the equation and the right side of the equation look the same.

HOWEVER, this must be done with extreme caution, for ONLY ONE SIDE OF THE EQUATION CAN BE CHANGED. If you ever even think of messing around with the other side of the equation, or even going so far as to CROSS-MULTIPLY, you will be promptly cast into the fearsome pit of trigonometric DOOM!!!!

In order to become a trig proof master, one must follow THE RULES (Actually, they're more like guidelines):

1. Don't lose track of what you are trying to prove. Randomly swapping in trig expressions for other trig expressions doesn't usually work. A better plan would be to look at what the end result should be before diving in. Verifying trig identities is not guess and check. (For an example of guess and check, see 'finding the roots of a polynomial.')

2. Work on making the more complicated side less complicated (or rewrite everything in terms of sine and cosine). Making something more complicated than it already is is generally a waste of time.

3. Work from minor functions (csc, sec, cot, tan) to major functions (sin and cos). This makes sense when you realize that sin and cos are a lot less complicated. (See rule #2)

4. Use the common denominator. When you find a common denominator, you can put two fractions together to form one fraction. And, believe it or not, one fraction is in fact less complicated than two fractions (See rule #2). I know, it's weird, but 1 really is less than 2.

5. Use conjugates. Remember how, in the first section of this post, sin θ was equal to 1 / csc θ? Guess what? It still is! And guess what else? In general, sin θ is a lot less complicated than 1 / csc θ. (Rule #2) Is this starting to sound familiar?

6. Use

Did you notice that the left hand side is one fraction and the right hand side is two? And did you notice that they all have a common denominator? Does anyone remember rule #4?

7. Use the Force... just checking to see if anyone was paying any attention.

8. Try not to take the square roots of things or raise them to powers. Remember how we're only supposed to change one side of the equation? If you suddenly make one side exponentially bigger or smaller, what are you supposed to do with the other side to make them match up?

9. Remember:

Last time I checked, trying to put two fractions together by finding a common numerator was not considered to be a very good idea. People who do this are getting way too overexcited about rule #4.

10. A trig proof is like an Oreo cookie. There is always more than one way to verify a trig identity. If your first attempt fails miserably, don't be afraid to go back and try something new!

11. DO NOT CROSS-MULTIPLY! DO NOT ASSUME THE LEFT HAND SIDE EQUALS THE RIGHT HAND SIDE! DO NOT MOVE STUFF FROM THE LEFT SIDE TO THE RIGHT SIDE! DO NOT MOVE STUFF FROM THE RIGHT SIDE TO THE LEFT SIDE! DO NOT PASS GO! DO NOT COLLECT $200! JUST LEAVE THE OTHER SIDE ALONE!

Now let's try asample identity: