
                         Review of Trig Identities.
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┌──────────────────┐
│ Basic Identities │
└──────────────────┘

               sin (-x)  =  - sin x

               cos (-x)  =  cos x

                  2       2
               sin x + cos x  =  1

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

      cos (x + y)  =  cos x cos y  -  sin x sin y
                                                      2         2
      sin 2x  =  2 sin x cos x           cos 2x  = cos x  -  sin x

There are many other useful identities which you must know about, for example
the following occur frequently in calculus problems :

         2             2                    2         2
      tan x + 1  =  sec x            1 + cot x  =  csc x

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

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

         2
      sin (x/2)  =  (1 - cos x)/2 

         2
      cos (x/2)  =  (1 + cos x)/2 

                       tan x  +  tan y
      tan (x + y)  =   ───────────────
                       1 - tan x tan y


Instructors aren't too happy with students who can't produce these with ease.
However, all of these can be obtained from the previous group very easily.
As an exercise you should make sure you can obtain all of them that way.
In general it is wise to remember a few formulae correctly and be able to
derive the rest from them.  RAM is very cheap, but until you can install it
in your brain, you are stuck with what you have.  Moreover, there is reason
to believe that it deteriorates with age!



┌─────────────────┐
│ Common Mistakes │
└─────────────────┘

WRONG       sin (2x) = 2 sin(x)      RIGHT    sin (2x) = 2 sin(x)cos(x)

with similar errors for   cos(2x), tan(2x), etc.,  for example

WRONG     3 tan(x/3) = tan(x)


WRONG  sin(x + h) = sin x + sin h

RIGHT  sin(x + h) = sin x cos h + cos x sin h

with similar errors for   cos(x + h), tan(x + h), etc.

             2  ___
WRONG     sin (\▌x )  =  sin (x)    The left-hand side cannot be simplified.


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