91Ó°ÊÓ

The arctangent function, denoted as \(\tan^{-1}(x)\), is an inverse trigonometric function. It tells us the angle whose tangent is \(x\). For instance, if \(\tan(\theta) = x\), then \(\theta = \tan^{-1}(x)\). This function is very useful in trigonometry when we need to determine an angle based on the value of its tangent. When you see an expression like \(\tan^{-1}\left( \frac{2x}{1-x^{2}} \right)\), it's a signal that we need to find out which angle's tangent corresponds to \(\frac{2x}{1-x^{2}}\). Remember, the range of \(\tan^{-1}(x)\) is typically between \(-\frac{\text{Ï€}}{2}\) and \( \frac{\text{Ï€}}{2}\), corresponding to every possible result we can get from the tangent function.

Trigonometric identities simplify expressions involving trigonometric functions, such as sine, cosine, and tangent. One important identity is the double-angle formula for tangent, which states: \[ \tan(2y) = \frac{2 \tan(y)}{1-\tan^2(y)} \] This is particularly useful in reducing complex expressions or solving equations. In the given exercise, we used this identity to solve for \(y\) by substituting \(y = \tan^{-1}\left( \frac{2x}{1-x^{2}} \right)\). With this substitution, we derive \[ \tan(y) = \frac{2x}{1-x^2} \] Applying the identity \[ \frac{2 \tan(y)}{1- \tan^2(y)} = \frac{2x}{1-x^2} \] helps us validate that \(y = \tan^{-1} \left( \frac{2x}{1-x^2} \right) \). Knowing how to use such identities efficiently makes solving trigonometric equations more manageable.

Solving trigonometric equations can often be broken down into simpler steps using trigonometric identities and inverse functions. In the provided problem, we start with the equation: \[ h(x) = \tan^{-1} \left( \frac{2x}{1-x^2} \right) \] By setting \(y = \tan^{-1} \left( \frac{2x}{1-x^2} \right)\), we get \[ \tan(y) = \frac{2x}{1-x^2} \] To solve for \(x\), we recognize that we can use the identity for \( \tan(2y)\), which helps us simplify the expression. Substituting back the functions, we ensure that our solution covers the entire defined domain. The verification step is crucial as it confirms that the solution consistently solves the equation over the valid input range. Thus, following these steps methodically ensures precision and accuracy in solving trigonometric equations.