Expert: Paul Klarreich - 12/12/2006

Question
use the product-to-sum forulas to write the product as a sum or differance
a) 3cos2xsin6x

use the sumto product formulas to find the exact value of hte expression
cos165-cos75

write teh following expression as a product
sin2x-cos6x

use the double angle formulas to find the exact value of sin2u and tan2u, given that cosu=2/7 3pi/2<u<2pi

verify the identity cos{x+pi/2}=-sin x

verify the identity cot^2x-cos^2x=cot^2xcos^2x

thanks alot if you can get any of these back to me that would be much appreciated.
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The text above is a follow-up to ...

-----Question-----
verify the identity: tan x cot x/cosx=sec x
verify the identity: 1+csc/sec-cot=cos
-----Answer-----
Questioner:  cody
Category:  Advanced Math
 
Subject:  pre-cal (trig)
Question:  verify the identity: tan x cot x/cosx=sec x
verify the identity: 1+csc/sec-cot=cos
.............................
Hi, Cody,

I think you mean to PROVE these identities.  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 as if this were an equation to be solved.  

The 'work' on each side uses:
A. Basic quotient identities:  tan = sin/cos, cot = cos/sin
B. Basic reciprocal identities: sec = 1/cos and v.v.,  csc = 1/sin and v.v., and
  cot = 1/tan and v.v.
B1. Alternative forms of those:  sec cos = 1;  csc sin = 1;  tan cot = 1.
C. Pythagorean identities:  sin^2 + cos^2 = 1 and variations;  sec^ = 1 + tan^2, etc.
D. Basic algebraic techniques, such as fractions.

Let's go:

tan x cot x
-----------  =  sec x
 cos x

   1
-----------  =  sec x   [Reciprocal]
 cos x

 sec x     =  sec x   [Reciprocal again]

Finito.
.....................
1+csc/sec-cot=cos    IS NOT PARENTHESIZED CAREFULLY.  If it is exactly as written, it

says:
   csc x
1 + ----- - cot x     =  cos x
   sec x
Which goes like this:

   1/sin x
1 + -------- - cot x  =  cos x
   1/cos x

   cos x
1 + ------ - cot x  =  cos x
   sin x

1 + cot x - cot x  =  cos x

         1    =  cos x

which is certainly not an identity.

Could it mean:

 1 + csc x
-------------  =  sec x
sec x - cot x

which does not work out, either.

OR:
      csc x
1 +  -------------  = sec x
    sec x - cot x

Likewise no good.

Check over the example and try again.


Answer
Questioner:  cody
Category:  Advanced Math
 
Subject:  Proving trig identities
Question:  

1. use the product-to-sum forulas to write the product as a sum or differance
  3 cos 2x sin 6x

2. use the sum-to-product formulas to find the exact value of the expression
 cos 165 - cos 75

3. write the following expression as a product
  sin2x-cos6x

4. use the double angle formulas to find the exact value of sin2u and tan2u, given that

cos u = 2/7, where 3pi/2 < u < 2pi  [u is in the fourth quadrant]

5. verify the identity cos{x+pi/2} = - sin x

6. verify the identity cot^2x-cos^2x=cot^2xcos^2x

thanks alot if you can get any of these back to me that would be much appreciated.

Hi, Cody,

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.

Look at:

http://www.sosmath.com/trig/Trig5/trig5/trig5.html

for a list of trigonometric identities that you can refer to.  Print that page and have it in front of you.  I will just refer to those and give you the clue as to how to proceed, since you are sending a LOT of questions here.  Generally, if you just can't do your entire homework assignment, you are taking a subject that is too difficult.

AND, I will insist on certain things from you.  If I send you a solution, it won't make much sense to you unless I put it in a readable format without typos or bad spellings, and I have to space it properly to make it all legible.  That is why, for example, I put in the WARNING that you see above.  

So I ask the same of you.  If you want to send along other questions, then be sure to:

A. Space them out nicely so they are readable.
B. Parenthesize carefully so they are properly interpreted.
C. Fix your typos before you hit 'send.'  If you typed 'teh' instead of 'the', take a few seconds to go back and fix it.  I do it, and so can you.

..................................

1. use the product-to-sum forulas to write the product as a sum or differance
  3 cos 2x sin 6x

the last formula in the list has:  cos u sin v.

Let  u = 2x  and  v = 6x  and apply that.

,,,,,,,,,,,,,,,,,,,,,,,,
2. use the sum-to-product formulas to find the exact value of the expression
 cos 165 - cos 75

The last formula in the Sum-to-product list has:  cos u - cos v

Let  u = 165  and  v = 75.  You will get angles of 120 degrees and 45 degrees, which you can evaluate exactly, using diagrams and radicals, like sqrt(2) or sqrt(3).


3. write the following expression as a product
  sin2x - cos6x

None of the S-to-P expressions is exactly like this, so use a Cofunction Identity to write  cos 6x = sin(pi/2 - 6x) and use the formula:
         u + v       u - v
sin u - sin v = 2 cos (-----) sin (-----)
         2          2
with u = 2x,  v = pi/2 - 6x.  Then:

u + v =   pi/2 - 4x
u - v = - pi/2 + 8x

sin u - sin v = 2 cos (pi/4 - 2x) sin (-pi/4 + 4x)

which is a product.


4. use the double angle formulas to find the exact value of sin2u and tan2u, given that cos u = 2/7, where 3pi/2 < u < 2pi  [u is in the fourth quadrant]

First, you will want to find that  sin u = -sqrt(43)/7  and tan u = -sqrt(43)/2  by making your standard 'quadrant diagram' and labeling the sides of your triangle.

Then:

sin 2u = 2 sin u cos u,  and use those values.
         2 tan u
tan 2u = ------------,  and use those values.
        1 - tan^2 u


5. verify the identity cos{x+pi/2} = - sin x

Use a sum-difference formula for cos(u + v), with  u = x, v = pi/2.  Then determine your values for  sin pi/2 and cos pi/2 from your knowledge of the graphs of sine and cosine.

6. verify the identity cot^2x-cos^2x=cot^2xcos^2x

  cot^2 x - cos^2 x        =  cot^2 x cos^2 x
  cos^2 x  
  -------- - cos^2 x       =  cot^2 x cos^2 x
  sin^2 x

cos^2 x    cos^2 x sin^2 x
-------- - ---------------- =  cot^2 x cos^2 x
sin^2 x       sin^2 x

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

cos^2 x cos^2 x
----------------          =  cot^2 x cos^2 x
   sin^2 x

cos^2 x cos^2 x          cos^2 x
----------------          =  -------- cos^2 x
   sin^2 x          sin^2 x  

cos^2 x cos^2 x          cos^2 x cos^2 x
----------------          =  ---------------
   sin^2 x          sin^2 x