AboutScott A Wilson Expertise I can answer any question in general math, algebra, complex mathematics, trigonometry, pre-calculus, probability, statistics, ... is there an end to this list?
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I worked at The Boeing Company for over 5 years.
Education/Credentials: MS degreee in Mathematics from Oregon State Univeristy; taken well over 100 hours of upper division credits in mathematical courses such as calculus, statistics, probabilty, linear algrebra, powers, linear regression, matrices, and more.
I graduated with honors.
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Expert: Scott A Wilson Date: 5/12/2008 Subject: Trig Identites
Question QUESTION: Hi, I'm having trouble solving this trig identity,
1-(sin^2xtanx/tanx+1)-(cos^2x/1+tanx)=sinxcosx
ANSWER: Is this really suppose to be
1-(sin^2(x)tan(x)/(tan(x)+1))-(cos^2x/(1+tan(x))=sin(x)cos(x)?
I think so!
Turn tan(x) into sin(x)/cos(x), don't forget parentheses, and combine the fractions. You will get a sin^3(x)+cos^3(x) on top.
Note the a^3+b^3=(a+b)(a^2-ab+b^2).
Thanks for this and previous questions this year.
---------- FOLLOW-UP ----------
QUESTION: I can't get sin^3(x)+cos^3(x) on top. It doesn't work. And once I do get sin^3+cos^3 on top how does that turn into sin(x)cos(x)?
Answer Take sin^2(x)tan(x)/(tan(x)+1).
It is known that tan(x)=sin(x)/cos(x), so this is
sin^2(x)(sin(x)/cos(x))/(sin(x)/cos(x) + 1).
Multiply top and bottom by cos and note the sin(x)'s combine to give
sin^3(x)/(sin(x)+cos(x)).
Take cos^2x/(1+tan(x)).
Multiply the top and bottom by cos(x). On top this gives cos^3(x) and on the bottom gives cos(x) + cos(x)tan(x). Again, it is known that tan(x)=sin(x)/cos(x), so the bottom is cos(x) + sin(x).
Our final fraction is cos^3(x)/(cos(x)+sin(x).
The fractions can be combined (since the denominators are equal, i.e., sin(x)+cos(x)=cos(x)+sin(x)), and both are being subtracted from 1) and as I gave you how to factor a^3+b^3, you can factor sin^3(x)+cos^3(x).
In other words, sin^3(x)+cos^3(x)=
(sin(x)+cos(x))(sin^2(x)-sin(x)cos(x)+cos^2(x)) and there is a sin(x)+cos(x) in the denominator, so the they cancel.
Note that all of this is subtracted from one, so we have
1 - sin^2(x) + sin(x)cos(x) - cos^2(x). This turns into
1 - (sin^2(x)+cos^2(x)) + sin(x)cos(x).
Recognize sin^2(x)+cos^2(x)? All you're left with is the answer.
