If \(\sin^2{x} - \cos^2{x} = 1\), find the value of \(x\).

Step 1 ：Recognize the equation \(\sin^2{x} - \cos^2{x} = 1\) as a difference of squares, which can be factored into \((\sin{x} - \cos{x})(\sin{x} + \cos{x}) = 1\)

Step 2 ：Knowing that both \(\sin{x}\) and \(\cos{x}\) ranges from -1 to 1, the only way that their product can be 1 is when both of them are -1 or 1

Step 3 ：Therefore, the possible solutions for the equation are when \(\sin{x} = \cos{x} = -1\) or \(\sin{x} = \cos{x} = 1\)

Step 4 ：However, it is impossible for \(\sin{x}\) and \(\cos{x}\) to be equal to 1 or -1 at the same time, because the maximum value of \(\sin{x}\) and \(\cos{x}\) is 1 and their minimum value is -1

Step 5 ：Hence, there are no solutions for the given equation