Expert: Paul Klarreich - 12/4/2008

Question
I am having problems solving a few trigonometric identities. If you could help, I would really appreciate it!

1. Prove the identity:

(1 + secx)/secx  = sin^2x/(1 - cosx)

2. Prove the identity:

sinx(cosx(x/2))-1 = cos x

3. Solve the identity and give all solutions:

(1 - cosx)/(1 + cosx) = 3

4. Solve the identity and give all solutions:

√3 tan3x + 3 = 0

**Only the first "3" is in the square root symbol

Answer
Questioner:   Tyler
Category:  Advanced Math
Private:  No
 
Subject:  Trigonometric Identities
Question:  I am having problems solving a few trigonometric identities. If you could help, I would really appreciate it!
...........................
Hi, Tyler,

Before I start, note: Since my recent difficulties I have been reducing my typing.  I use a lot of abbreviations:

s  = sin x
c  = cos x
s2 = sqrt(2)
s3 = sqrt(3)


1. Prove the identity:

(1 + secx)/secx  = sin^2x/(1 - cosx)

Try changing everything to sines and cosines:

1 + 1/c     s^2
-------- = -----
 1/c      1 - c

Now some algebra: Clear fractions on the left:

c + 1       s^2
-------- = -----
 1        1 - c

Use a Pythag identity:

c + 1      1 - c^2
-------- = -------
 1        1 - c

Factor, and .... I think you can handle it from here.




2. Prove the identity:

sinx(cosx(x/2))-1 = cos x


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3. Solve the identity and give all solutions:

NO, NO, NO --- you don't solve an identity.  If it is an identity, then everything is a solution.  You mean to solve the EQUATION.  Always, always, be careful of the vocabulary.  

(1 - cosx)/(1 + cosx) = 3

Try cross-multiplying?

1 - c = 3 + 3c
- 2 = 4c

c = - 1/2

now x is in quadrants 2,3, and the reference angle is 60 degrees.  Take it from there.


4. Solve the identity and give all solutions:

sqrt(3) tan(3x) + 3 = 0

sqrt(3) tan(3x) = - 3


tan (3x) = - sqrt(3)

Now 3x is in quadrants 2,4, and the reference angle is 60 degrees.

So 3x = 120 or 300.  Your turn.