91Ó°ÊÓ

The inverse tangent function, denoted as \( \tan^{-1}(x) \), is a crucial component in trigonometry. It's the inverse operation of the regular tangent function. Essentially, if \( y = \tan(x) \), then \( x = \tan^{-1}(y) \). This inverse function helps find the angle when the tangent value is known. It plays an important role in many applications, such as solving triangles and in calculus. Remember, the principal value range for \( \tan^{-1}(x) \) is between \(-\frac{\pi}{2}\) and \(\frac{\pi}{2}\). This ensures that each tangent value corresponds to exactly one angle. When using \( \tan^{-1}(x) \), always consider this range to find the correct angle.

The inverse cotangent function is represented as \( \cot^{-1}(x) \). Similar to inverse tangent, it reverses the cotangent function: if \( y = \cot(x) \), then \( x = \cot^{-1}(y) \). In terms of its principal range, \( \cot^{-1}(x) \) is found between \(0\) and \(\pi\). This allows for a unique angle corresponding to any cotangent value. This function is particularly useful when working with angles and trigonometric identities in problems. It's important to note that, unlike tangent, cotangent is undefined at multiples of \(\pi\) where sine equals zero, hence the principal range choice of \(0\) to \(\pi\). Understanding \( \cot^{-1}(x) \) facilitates solving equations involving cotangent and its related trigonometric identities.

Trigonometric identities are foundational to solving equations like the one presented in the exercise. These identities help in transforming expressions into more manageable or comparable forms. For instance, the identity \( \sin(\tan^{-1}(x)) = \frac{x}{\sqrt{1+x^2}} \) comes from forming a right triangle where the opposite side is \(x\) and the adjacent side is \(1\). Similarly, for cosine, \( \cos(\cot^{-1}(x)) = \frac{1}{\sqrt{1+x^2}} \), you imagine a right triangle where the adjacent is \(x\) and the opposite is \(1\). Such identities stem from rewriting inverse trigonometric expressions in terms of basic sine, cosine, and tangent functions. They are indispensable tools for verifying equations and transitioning between different trigonometric forms.