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

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Accepted Answer

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 1Step: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 -1Step:5 ：Hence, there are no solutions for the given equation

Answer

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 1Step: 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 -1Step: 5 ：Hence, there are no solutions for the given equation

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