1. ## Simplifying

How can I simplify the following to sec^2(x):

tan(sqrt(x))/(sqrt(x)) + (2sqrt(x)/(2sqrt(x)*cos^2(x)) - tan(sqrt(x))/(sqrt(x))

That simplifies to the sec^2(x), according to a TI 89 calculator, but not sure exactly how.

And then, for:

x/(x^1) - x/(x^2+1)^2, for some odd reason I get x/(x^2+1)^2 when I try to simplify it, but it should be x^3/(x^2+1)^2.

2. Originally Posted by PhilipJ
How can I simplify the following to sec^2(x):

tan(sqrt(x))/(sqrt(x)) + (2sqrt(x)/(2sqrt(x)*cos^2(x)) - tan(sqrt(x))/(sqrt(x))

That simplifies to the sec^2(x), according to a TI 89 calculator, but not sure exactly how.

And then, for:

x/(x^1) - x/(x^2+1)^2, for some odd reason I get x/(x^2+1)^2 when I try to simplify it, but it should be x^3/(x^2+1)^2.

The first one works out a little too easily. I'm suspecting a typo? Either way, it does work. Note that the first and last term are identical, except for a sign. So they add to zero. That leaves the second term, in which you can cancel a common 2 sqrtx, leaving 1/(cos x)^2. Since sec x = 1/cos x, we get (sec x)^2.

Not sure what to make of your second problem.
x/(x^1) - x/(x^2+1)^2
If you typed it correctly, the problem is wrong. x/(x^1) = 1.

-Dan

3. Wow, I am an idiot. I was doing multiplication of the last term when trying to simplify the first one. No wonder I couldn't get it to work!

And on the second one, it is indeed a typo.

I had: x/(x^1) - x/(x^2+1)^2. It should be as follows:

x/(x^2+1) - x/(x^2+1)^2

4. Originally Posted by PhilipJ
Wow, I am an idiot. I was doing multiplication of the last term when trying to simplify the first one. No wonder I couldn't get it to work!

And on the second one, it is indeed a typo.

I had: x/(x^1) - x/(x^2+1)^2. It should be as follows:

x/(x^2+1) - x/(x^2+1)^2
Then we get:
$\displaystyle \frac{x}{x^2+1}-\frac{x}{(x^2+1)^2}$

$\displaystyle =\frac{x}{x^2+1} \, \frac{x^2+1}{x^2+1} - \frac{x}{(x^2+1)^2}$

$\displaystyle =\frac{x(x^2+1)}{(x^2+1)^2}-\frac{x}{(x^2+1)^2}$

$\displaystyle =\frac{x^3+x}{(x^2+1)^2}-\frac{x}{(x^2+1)^2}$

$\displaystyle = \frac{x^3+x-x}{(x^2+1)^2}$

$\displaystyle =\frac{x^3}{(x^2+1)^2}$

-Dan