RE:
(- sin (x/2) /( 2 sin (x/2) + cos (x/2))
is an alternate representation for,
1 / ( 4 tan (x/2) + 2 )
Thanks for your help, sorry I posted 3 times, I thought you didn't understand what I needed. I really do appreciate your time spent.
just for your FYI, if you are interested, see towards the bottom of the link, before the plot, where it says they are equivalent. I guess Wolframe Alpha is wrong... huh, I couldn't quite get it and I'm sure you know more than I.
Arg, It won't let me post a link. Oh well, sorry and thanks again.
I'm new to this. I didn't know you could reply to the original post? (testing with this reply), otherwise I would not have kept re-posting as a 'new' question.
I just noticed some replies as I was reading other questions. again, my apologies. I feel like I should change my name if I have another question! have a good night.
Carol
Since we did not resolve your question, I am curious where you got this problem.
Is it from a textbook?
Is in an assignment problem?
Just curious, since it obviously not a true statement.
It's actually the end of an integration problem from a college calc text I have from 1971. The problem involved Weierstrass Substitution. The part I posted was after converting from the u-substition to an answer in terms of x.
The orig prob. is
integrate (cos x )/(3 cos^2 x - 5 cos x - 4 sin x cos x ) dx
I'll paste the part I was referring to in a new reply post.
Continued, also this isn't an official assignment. I'm teaching myself calculus. I just love math. I took cal 30 thirty ago but don't remember much.
I use Wolframe Alpha to check my answers.
here's the web paste without the beg. address part, you'll have to copy and paste. look at the bottom just before the graph.
wolframalpha dotcom period slash input/?i=integrate+%28cos+x++%29%2f%283+cos^2+x+-+5+cos+x+-+4+sin+x+cos+x+%29++dx&asynchronous=pod&s=29&incTime=true
I did manage to cut and paste your link, here is what I found:
http://www.wolframalpha.com./input/?i=integrate+%28cos+x++%29%2f%283+cos^2+x+-+5+cos+x+-+4+sin+x+cos+x+%29++dx&asynchronous=pod&s=29&incTime=true
I did notice that the equivalence was between
-sin(x/2)/(2sin(x/2) + cos(x/2)) and
- 1/(cot(x/2) + 2)
which is different from what you had.
as I said it my first reply to an earlier post, since all angles are x/2 , for convenience I will use Ø for x/2
so let's prove
-sinØ/(2sinØ + cosØ) = -1/(cotØ + 2)
RS = -1/(cosØ/sinØ + 2)
= -1/((cosØ + 2sinØ)/sinØ)
= =sinØ/(2sinØ + cosØ) , I multiplied by the reciprocal
= LS
OK then!!!! We got it.
Btw, good for you to work on this all by yourself.
Looks like you have reached quite an advanced stage already.
sorry for the late thank you!!
I lost the post and couldn't find it.
doubt you will see this, and I didn't want to clog up the page with a new post.
you're the greatest!!
To confirm that the expressions (- sin(x/2) /(2 sin(x/2) + cos(x/2)) and 1 / (4 tan(x/2) + 2) are equivalent, we can simplify both expressions and compare the results.
Let's start with the expression (- sin(x/2) /(2 sin(x/2) + cos(x/2)):
Step 1: Simplify by factoring out sin(x/2):
(- sin(x/2) /(2 sin(x/2) + cos(x/2))) = -sin(x/2) / sin(x/2) * (1 / (2 + cos(x/2)))
Step 2: Cancel out sin(x/2):
-1 / (2 + cos(x/2))
Now let's simplify the expression 1 / (4 tan(x/2) + 2):
Step 1: Simplify by factoring out 2:
1 / (4 tan(x/2) + 2) = 1 / 2 * (1 / (2 tan(x/2) + 1))
Step 2: Rewrite tan(x/2) as sin(x/2) / cos(x/2):
1 / 2 * (1 / (2 (sin(x/2) / cos(x/2)) + 1))
Step 3: Simplify the expression by multiplying both the numerator and denominator by cos(x/2):
1 / 2 * (cos(x/2) / (2sin(x/2) + cos(x/2)))
Step 4: Simplify further by factoring out cos(x/2):
1 / 2 * (cos(x/2) / cos(x/2) * (1 / (2 sin(x/2)/cos(x/2) + 1)))
Step 5: Cancel out cos(x/2):
1 / 2 * (1 / (2 sin(x/2)/cos(x/2) + 1))
The final simplified expression matches the expression from the other representation, (-sin(x/2) / (2 + cos(x/2))). Therefore, we can conclude that both expressions are indeed equivalent.
Regarding the link you mentioned, I apologize for not being able to view it directly. However, based on the steps of simplification explained above, we can confidently say that the two expressions are equivalent.