Trigonometry Identities

Date: 07/30/97 at 14:14:18
From: luis manuel lozano
Subject: Trigonometry identities

 have a problem with these identities. I don't know the answer - 
please help me:

1.  1-tanA + secA = 1+tanA
    ------   ----   --------
     secA    tanA   secAtanA

2.  tanA - senA = secA+cosA
    -----  ----   ---------
    cscA - cotA   cscA+cotA

3.  tanA+sec3A-secA = tan2A+senA
    ---------------
         secA

4.  cosA+senAcotA = 2 senA
    -------------
        cotA

5.  senX   = 1+cosX
    ------   ------
    1-cosX    senX

6.  sen2X = cosX+1
    ------  ------
    secX-1   secX

7.  csc4Y-1 = csc2Y+1
    -------
    cot2Y

8.  1-3senY - 4sen2Y  = 1-4senY
    -----------------   --------
          cos2Y          1-senY

Thank you very much.

Date: 08/01/97 at 13:52:12
From: Doctor Rob
Subject: Re: Trigonometry identities

One technique that works very well in these examples is to express
everything in sight in terms of sin(A) and cos(A) (or X or Y instead
of A).  Another simplifying thing to do is to clear all fractions by
multiplying both sides of the equation by the least common multiple
of all its denominators. For example, I'll work through number 1 for 
you, and you do the rest:

   1 - tan(A) + sec(A) =  1 + tan(A)
   ----------   ------   -------------
     sec(A)     tan(A)   sec(A)*tan(A)

Clear fractions by multiplying by sec(A)*tan(A):

        tan(A) - tan^2(A) + sec^2(A) = 1 + tan(A).

Subtract tan(A) from both sides:

                -tan^2(A) + sec^2(A) = 1.

Write in terms of sine and cosine:

   -(sin(A)/cos(A))^2 + (1/cos(A))^2 = 1.

Clear fractions again by multiplying by cos^2(A):

                       -sin^2(A) + 1 = cos^2(A).

This should look familiar to you! Add sin^2(A) to both sides:

                                   1 = sin^2(A) + cos^2(A).

This is one of the fundamental identities, so you are done.

-Doctor Rob,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/