Verifying Trig Identities

*Verifying trig identities are like cleaning up a messy mess. You just “clean up” the messy parts till you get the answer. *

Lesson                                                        

Let say you were ask to verify this messy identity:

The goal is to change one side of the identity so that it matches the other side. 

        Therefore, we need to clean up this mess, , to get this clean, and nice mess, . Keep in mind that we can also expand this side, , to get the other side, . As long as both side look the same after verifying, we're good.

        Of course, before we begin cleaning up, we must know which part to clean up. The Fundamental Trigonometry Table can show you which part of the mess you can manipulate.  
      **  Link for the table: 
https://docs.google.com/open?id=0B5F97NIUTL8vRHVrcXJlQ1NJaVk**
        With the fundamental identities, you can just use simple substitution to change the” pieces”. For instance:
             In , can be substitute in for . 

            In , can be substitute in for .



There is no one way, to clean up a mess, or verifying trig identities, but you can try to always follow these tips:

1)  Usually start with the complicate side first. Try simplifying it so that it matches the other side. If that fail, STOP, and start on the other side. Remember, work on 1 side at a time only.

             

                **Start with the right side first, 

               simplify it to get the left side. It’s easier, why not?

         2) When see anything squared like or , think of Pythagorean Identities        

                **See cot^2(x) and csc^2(x)? Think of using the       Pythagorean Identities. 

         3) It’s common to change tan(x), cot(x), sec(x), and csc(x) to quotient form like                         sin(x)/cos(x), cos(x)/sin(x), 1/cos(x), 1/sin(x), respectively.

            **If you change cotx to cosx/sinx,you'll have more more                                                            options to verify the identity, why not do it?

   4) Where applicable, you also use various methods you learned in algebra, such as putting things over a common denominator, factoring out a common factor, like term, distribute, and so on.

  ** Factoring

                                           ** Least Common Denominator

                         ** Combining Like Terms

** Distributing

Examples                                                          

1) Verify

**The right side looks a bit more complicated. Let's start with the it.
 

**Using our quotient identity, we can change tanx to sinx/cosx

**After multiplying, the cosx cancel each other out, we are left with sinx equal sinx, the identity is VERIFIED!





 2) Verify  


     **The right side is more complicated, start with it.     



**Using the quotient identity, we can change tanx and  cotx to sinx/cosx, and cosx/sinx, respectively.

** Ahh LCD, instead of adding the denominator together, we could multiply the whole thing by sinxcosx over sinxcosx to get rid of the ugly fraction.

** Simplify


** Cos^2x, Ahhhh, Pythagorean theorems.                               Sin^2x + Cos^2x = 1 so we can change the denominator to 1.

 

 

** Stuck?? Since the left side have secx, we should try to get secx. We know that cosx = 1/secx, so there we go, change the cosx.


**After simplifying, we are left with sinx/secx = sinx/secx, this identity is VERIFIED.

**** REMEMBER*****, when verifying a trig identity, be SUPER formal, neat, and always work only on one side - like we did in these examples. Have Fun Verifying!

“We believe ‘The more you try, the more you succeed’” – TrigInADay

Here is some practice problems:
        https://docs.google.com/open?id=0B5F97NIUTL8vMlgwZ0l0cTlpcXM
Try it out! If you have any question, please don't hesitate to contact us at triginaday@gmail.com. We are more than happy to walk you through the problems!

-TrigInADay