What is the use of simplifying a question in math?

[pairofstrings]

Hello. I have been looking into trigonometric identities that helps to simplify a question in mathematics. But the question that I have is, why do I need to simplify a question?

For instance:
Sin(x) * Cos(x) means Sin(2*x)/2.
I have no answer that tells me why I simplify Sin(x) * Cos(x) to its corresponding answer.

I have plotted graphs of Sin(x) * Cos(x) and Sin(2*x)/2 and both the expression give the same result.

I request you to write all the words associated with this question.

Thank you.

 

[mfb]

Double-angle formula

Simplifying the result makes it easier to understand it, it makes comparisons easier, and it is easier to use it in further steps if it is not the final answer.
The question "what is the nicest answer" is not always easy to answer. It is obvious that $\displaystyle \left(\frac {3\sin(x)}{2} - \frac {\sin^2(x)}{2\sin(x)}\right) \left(\frac {4\cos^2(x)}{3\cos(x)} - \frac {\cos(x)}{3}\right)$ is not a proper final answer, although it is identical to your expression, but sin(x) cos(x) versus sin(2x)/2? I would prefer sin(x) cos(x) over sin(2x)/2 in most cases, unless the double angle (2x) appears elsewhere as well.

 

[rock.freak667]

In my experience it is better to put it in the simplest form for ease of future solutions or graph plotting.

For example say you are asked to plot the long trig expression mfb posted within a certain range and you don't have a plotter. So you are to use excel or some similar spreadsheet program. And if you simplified the expression, it might be something like (it is not though) sin(9x)cos(3x).

In excel, I'd find it easier to plot sin(9x)cos(3x) than the long expression where I am likely to make a mistake and thus get the final solution wrong.

 

[jedishrfu]

Looking at the sin cos expression its hard to picture its graph but looking at sin 2x I know immediately that it will look like a sin x graph but with a doubled cycle in the domain of 0 to 2pi

Its the same reason for recasting a linear equation into y = mx + b form so that you can get the y intercept and the slope immediately and can draw the graph.