a) Show that for small values of x, xtanx is approximately equal to x^2 and 2 - coshx is approximately equal to 1 - 0.5(x^2). Draw a conclusion from this regarding the probable number and approximate locations of roots on the interval [-1,1].
b) Use Newton's method to identify one of these roots to three decimal places.
xtanx = 2 - coshx
The Attempt at a Solution
My main problem is with the first part.
How does xtanx approximate to x^2 for small values of x? Better yet, how would I show that? Testing out 10^-10, xtanx = 1.745 x 10^-22 and x^2 = 10^-20, which is still 2 orders of magnitude off. Based on the second part of the question, this approximation should be fairly accurate for x's in [-1,1], but I wouldn't really say it is.
Even if this difference in order of magnitude is acceptable and still considered approximately equal, how exactly would I show that? (besides subbing in numbers and doing a comparison chart or something).
I'm assuming once I accurately show that these approximations are true, I can estimate the number and approximate locations of the roots of the equation by finding how many times x^2 intersects 1 - 0.5(x^2) then use the substituted equation in Newton's method to approximate the actual roots.