Expert: Paul Klarreich - 12/14/2006

Question
sec (csc - 1) = Cot / 1 + sin

Answer
Questioner:  Marie
Category:  Advanced Math
 
Subject:  Verify Idenity
Question:  sec (csc - 1) = Cot / 1 + sin
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Hi, Marie,

I think you mean to PROVE this identity.  The proper technique is to write the identity and then, working on one side at a time, reduce the two sides to the identical expression (hence the term 'identity').  Sometimes you do all the work on one side, sometimes you do some work on each side, but you do NOT do things like add, subtract, multiply, divide on both sides as if this were an equation to be solved.  

Also, I must ask you to be a good girl and

1. Write  "sec x",  not "sec", and "sin x", not "sin".  Otherwise,  sec(csc - 1)  makes it look as if you are taking the secant of (csc - 1).  I don't think that's what you meant.

2, ALWAYS PARENTHESIZE CLEARLY so you don't send something that will be misinterpreted.  You wrote:  cot / 1 + sin.

Does that mean  cot x + sin x?  I don't think you meant that.  I think you meant to parenthesize the "1 + sin x" and send this nice example:

WARNING: THE FOLLOWING DISCUSSION MAY CONTAIN FRACTIONS AND OTHER MATERIAL INAPPROPRIATE FOR CERTAIN COMPUTING SYSTEMS.  BE SURE TO VIEW IT IN A FIXED-SIZE FONT, SUCH AS COURIER.


Prove the identity:
                             cot x
    sec x (csc x - 1)  =   ----------
                           1 + sin x  

OK.  Nuff said.  We can go to work.  

When I see  "1 + sin x" I get the overwhelming urge to   'rationalize' by multiplying top and bottom by  (1 - sin x)  because I know that will give me  1 - sin^2 x  which I will change to  cos^2 x.

                             cot x    (1 - sin x)
    sec x (csc x - 1)  =   -----------------------
                           (1 + sin x)(1 - sin x)

                             cot x (1 - sin x)
    sec x (csc x - 1)  =   -----------------------
                               1 - sin^2 x

                             cot x (1 - sin x)
    sec x (csc x - 1)  =   -----------------------
                                cos^2 x

Now I like to change everything that ISN'T sine or cosine to those, using quotient identities or reciprocal identities:

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

Now I'll do a little algebra, like cancelling and combining fractions.

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

A teeny bit of simplification:(and writing the sin x and cos x factors in alphabetical order:

      1 - sin x        1 - sin x
    -----------   =   ------------
    cos x sin x        cos x sin x

and we are done.