Expert: Paul Klarreich - 5/6/2007

Question
Thanks for answering my other question.  I just have one more.  I need to find the exact value of
(1/cos^2 40deg.)-(1/cot^2 40deg).  I'm not sure how to go about solving this one either.

Answer
Questioner:   Sarah
Category:  Advanced Math

Question:  Thanks for answering my other question.  I just have one more.  I need to find the exact value of
(1/cos^2 40deg.)-(1/cot^2 40deg).  I'm not sure how to go about solving this one either.
............................................
Hi, Sarah,

****** VIEW THIS IN COURIER FONT *******

We will use these trig properties:
         1
tan x = -----    (I)
       cot x
       sin x
tan x = ------   (II)
       cos x

1 - sin^2(x) = cos^2(x)   (III)

As you are about to see, the example has nothing to do with 40 degrees.

   1            1
--------- -  ---------   >> The example.
cos^2(40)    cot^2(40)

    1   
--------- -  tan^2(40)   >> Use I
cos^2(40)

    1       sin^2(40)
--------- -  ---------   >>  Use II
cos^2(40)    cos^2(40)

1 - sin^2(40)
-------------   >> Combine fractions.
cos^2(40)

cos^2(40)
----------  >> Use III
cos^2(40)

= 1   

(You knew that would happen, didn't you?)