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Trigonometric equations can often look intimidating, but with a methodological approach, they can be dissected and solved with ease. To solve trigonometric equations effectively, one must be comfortable with the properties and inverses of trigonometric functions.

When tackling such equations, the goal is to isolate the trigonometric function of the variable you are solving for, much like you would solve for a variable in an algebraic equation. The process usually involves using inverse trigonometric functions to 'undo' the effect of the trigonometric functions applied to the variable. This typically requires knowledge of the core trigonometric identities and the unit circle.

Key steps include identifying the angles that correspond to given trigonometric values, using algebraic manipulation to simplify the equation, and being aware of the domain and range of the functions involved to ensure that the solutions are appropriate.

The inverse sine function, denoted as \(\sin^{-1}\) or \(\arcsin\), is the inverse operation of the sine function. It takes a number between -1 and 1 and outputs an angle between \( -\frac{\pi}{2} \) and \( \frac{\pi}{2} \).

An essential aspect to note about \(\sin^{-1}\) is that, because sine is not a one-to-one function over its entire domain, \(\sin^{-1}\) is defined only for a certain range of angles so that it can act as a proper function - meaning for each input, there is only one output.

How to Use \(\sin^{-1}\)

When applying \(\sin^{-1}\) to solve an equation, you're essentially asking for which angle in the restricted \( -\frac{\pi}{2} \leq \theta \leq \frac{\pi}{2} \) region, the sine value equals the number we have. In the context of the given exercise, the inverse sine of \(\frac{1}{\sqrt{5}}\) corresponds to an angle where the sine value is that fraction. Knowledge of the unit circle tells us that this angle is \( \frac{\pi}{4} \).

Similarly, the inverse cosine function, specified as \(\cos^{-1}\) or \(\arccos\), reverts the process of the cosine function. It accepts a value within the range of -1 and 1 and gives back an angle within the range \(0 \) and \(\pi\).

When solving trigonometric equations that involve \(\cos^{-1}\), understanding the associated angle is key. The inverse cosine gives us the angle whose cosine is the specified number. In the case of the provided exercise, we have \(\cos^{-1}x = 0\) which informs us that we need to find an angle whose cosine is zero. Following the properties of the cosine function and recall of the unit circle, we know that the angle whose cosine is zero is \(\frac{\pi}{2}\), but since we have the equation \(\cos^{-1}x = 0\), this tells us the angle is actually 0. Therefore, \(x \) is equal to the cosine of 0, which is 1.