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Understanding trigonometric identities is crucial to mastering trigonometry problem-solving. One such fundamental identity is the tangent addition formula which is given by
\[\tan(A + B) = \frac{\tan(A) + \tan(B)}{1 - \tan(A)\tan(B)}\].
This identity allows us to find the tangent of the sum of two angles if we know the tangents of the individual angles. By rearranging the terms and applying algebraic operations, we can often simplify complex trigonometric expressions into a more manageable form. Trigonometric identities are like the building blocks which, when combined appropriately, help to unravel the mysteries hidden within trigonometry equations.

The inverse tangent function, often denoted as arctan or \(\tan^{-1}\), is essential when we wish to find an angle whose tangent value is known. Unlike the regular tangent function, which takes an angle and gives us a ratio, the inverse tangent function takes a ratio and returns the corresponding angle. For instance, if we know that \(\tan(\theta) = 1\), the inverse function allows us to conclude that \(\theta = \frac{\pi}{4}\) in radians, since that's the angle at which the tangent equals one within the principal range \(-\frac{\pi}{2}\) to \(\frac{\pi}{2}\). This function is particularly helpful when reverse-engineering trigonometry problems to find unknown angles.

Trigonometry problem-solving often involves a series of steps where strategic use of identities, inverse functions, and algebraic manipulation is imperative.

The textbook exercise provided is a perfect illustration of this: starting with a trigonometric identity (the tangent addition formula), moving to substitute known values, simplifying the expression, and finally using the inverse tangent function to find the sum of the angles. This process not only reinforces the understanding of various trigonometric concepts but also enhances problem-solving skills. When approaching such problems, it's vital to work systematically and pay careful attention to the mathematical relationships present in the identities used, as these are the keys that will unlock the solution.