[talanum1 (OP)]

We cannot derive the above by algebraic equation manipulation, for if you take the negative square root on both sides you get:

-sqrt(x^2) = - sqrt(1)

and the negative sign cancells both sides, so this doesn't work. Taking the negative square root on one side and positive on the other is not doing the same thing on both sides and is illegal.

Substituting -1 in LS is not equation manipulation.

[Bored chemist]

The square root function isn't a well behaved function in maths.
It gives two values.

[chiralSPO]

x² = 1
x² – 1 = 0
(x+1)(x–1) = 0
x = 1 and x = –1 both satisfy the last equation

[evan_au]

You can also solve it graphically, by graphing y = x²-1

Then you look to see where this graph crosses the x-axis, ie y = 0 = x²-1

There are two such locations where this graph crosses the x-axis, at x = 1 and x = -1

Parabola_Graph.png (8.65 kB . 472x274 - viewed 2373 times)

[evan_au]

You can also solve it numerically, using the Newton-Raphson method.
See: https://en.wikipedia.org/wiki/Newton%27s_method#Square_root
To find the negative root, start with a negative initial estimate

[evan_au]

Here is the results of solving this equation numerically, with starting guesses of:
- x = +4.5: converges to the root at x = 1 in 6 iterations (green line)
- x = -4: converges to the root at x = -1 in 6 iterations (orange line)
- Using 12-digit floating-point numbers

Starting with the initial guess, you follow the tangent of the curve to find a better guess.

Newtons_Method.png (13.74 kB . 475x275 - viewed 2427 times)

[vhfpmr]

The square root function isn't a well behaved function in maths.
It gives two values.

Inverse trig functions have an infinite number of solutions.

[evan_au]

In mathematics, a function is a binary relation between two sets that associates every element of the first set to exactly one element of the second set.

The definition of a mathematical function is that it has exactly one value at every point where it is defined.

So when you have something like Square Root or Inverse Sine where there are multiple values, mathematicians arbitrarily select one of these values as the "Principal Value" of the function.
- For Square Root (domain x ≥ 0), they arbitrarily choose the positive square root y ≥ 0 (even though -y is also a valid answer)
- For Inverse Sine (domain -1 ≤ x ≤ +1), they arbitrarily choose the range -π ≤ y ≤ +π  (even though any multiples of 2π radians y ± 2nπ are also valid answers)
- They just expect that the user is aware that there are other answers that should be considered

This definition fits well with calculators and computers, with one number into a function, and one number out of the function
- Computer software does not cope well if one computer returns answers to sin^-1 with a range -π ≤ y ≤ +π, and another computer returns values with a range +π ≤ y ≤ +3π.
- Computers don't generally cope well with functions producing an infinite list of answers - use a symbolic maths package for this.

The definition of a mathematical function as being single-valued can be generalized to a multi-valued function, see discussion of square root (and the complexities of doing this in the general case):
See: https://en.wikipedia.org/wiki/Function_(mathematics)#Multi-valued_functions