Expert: Josh - 11/7/2004

Question
Can you please explain this part:Expanding the left hand side of the original expression,
(sinx+cosx)^2-1 =[sin(x)]^2+ 2*sin(x)*cos(x)+[cos(x)]^2 -1
         =([sin(x)]^2+[cos(x)]^2-1)+ 2*sin(x)*cos(x)
         =(1-1)+2*sin(x)*cos(x)  ...[**]
         =sin(2x) using [#1] with a=b=x.
I don't understand what you did to get sin2x.Please explain.Thanks.
-------------------------
Followup To
Question -
The expression(sinx+cosx)^2-1 is equivalent to sin 2x.Why?
Answer -
Hi Jeff,

I can kind of understand where you are coming from.
Mathematicians have been ridiculed since the dawn of time, because they "like proofs". In a sense, that is true because the only way to show that an expression (or claim) is true (which isn't immediately obvious) is by proving it using established facts.

High school and college students (except inspired mathematicians) hate them for a number of reasons.
(i) The proofs are often unappealing - the steps used to arrive at the conclusion are contrieved and even less obvious than the thing we are trying to prove, so it is easy to lose the plot.
(ii) While the formal arguments are sound, they are often not very intuitive to the average person. There is so much left unsaid (which would be helpful if someone actually explains what is going on) but is not actually required in a proof.

With that said, let me draw you some pictures to see what this expression is all about. Refer to the figures at www.geocities.com/joshcameron_ae/Archive/jeff.html

INTRODUCTION: Figure 1 shows that the sine wave and cosine wave have the same shape. They have identical values in principle, except, the sine wave is delayed or shifted to the right by 90 degrees.

CLAIM (read this carefully and refer to diagrams at www.geocities.com/joshcameron_ae/Archive/jeff.html):
The expression [sin(x)+cos(x)]^2 -1 = sin(2x)  argues that by adding (superimposing) sin(x) with the cos(x) (see red line in Figure 2), then squaring its values ([sin(x)+cos(x)]^2 is plotted as a black line in Figure 2), the resulting waveform has the same amplitude and periodicity as the sine wave with twice the frequency, namely, sin(2x).

To obtain exactly sin(2x), we only need to shift the [sin(x)+cos(x)]^2 waveform by one unit down in the vertical direction. This is all the "-1" part is saying.

IMPORTANT PHYSICAL PRINCIPLE: The underlying concept of this expression is that we can increase (double) the frequency of an oscillating waveform, by superimposing the sine wave and cosine wave with half the desired frequency. Remember that sin(x)=cos(x-90) lags behind cos(x) by 90 degrees, that's all to it.
You can repeat this argument to get [sin(2x)+cos(2x)]^2-1 = sin(4x) for instance.

FINALLY, HAVING SEEN THE PICTURES, HERE IS THE MATHEMATICAL PROOF,

Established fact: sin(a+b)=sin(a)cos(b)+cos(a)sin(b)...[#1]

Expanding the left hand side of the original expression,
(sinx+cosx)^2-1 =[sin(x)]^2+ 2*sin(x)*cos(x)+[cos(x)]^2 -1
         =([sin(x)]^2+[cos(x)]^2-1)+ 2*sin(x)*cos(x)
         =(1-1)+2*sin(x)*cos(x)  ...[**]
         =sin(2x) using [#1] with a=b=x.

Let me know if you have further questions.

Answer
Hi Jeff,

Please refer to
http://www.geocities.com/joshcameron_ae/Archive/Trigonometry.pdf
for further clarification.

Don't hesitate to ask again if you have any other question.

Cheers.
==============================================
You will need an Adobe reader to view for this[http://www.adobe.com/]