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Inverse trigonometric functions are the reverse processes of the regular trigonometric functions. They allow us to determine an angle when the trigonometric ratio is known. For example, if we know the value of tangent for a certain angle, using the inverse tangent function, denoted as \( \tan^{-1} \), we can find this angle itself.
The inverse cotangent function, \( \cot^{-1} x \), is used to find the angle whose cotangent is \( x \). This function is particularly useful in solving equations or inequalities involving cotangent. It's important to remember that just like other inverse trigonometric functions, \( \cot^{-1} \) has specific domain and range restrictions to ensure precision:

• The domain of \( \cot^{-1} \) includes all real numbers.
• The range of \( \cot^{-1} \) is \((0, \pi)\), since cotangent is undefined at 0 and \( \pi \).

Using inverse trigonometric functions helps in breaking down and simplifying complex mathematical problems, especially in calculus and coordinate geometry.

The cotangent function, denoted as \( \cot(x) \), is one of the six fundamental trigonometric functions. It is defined as the reciprocal of the tangent function. Mathematically, this is expressed as \( \cot(x) = \frac{1}{\tan(x)} \) or \( \cot(x) = \frac{\cos(x)}{\sin(x)} \).
Understanding cotangent is essential, especially when dealing with right-angled triangles or periodic phenomena in waves and oscillations. It gives us an additional ratio to work with beyond sine, cosine, and tangent, allowing for more flexibility in problem-solving. Here are a few key points:

• Cotangent has asymptotes, or undefined values, wherever the sine of the angle is zero, such as at \( x = n\pi \) where \( n \) is an integer.
• The cotangent function has a period of \( \pi \), meaning the function repeats its values every \( \pi \).

Due to these properties, the cotangent function plays a critical role in higher-level math problems, including those found in competitive exams like the IIT JEE.

IIT JEE, short for the Indian Institutes of Technology Joint Entrance Examination, is one of the most challenging entrance exams for engineering aspirants in India. The mathematics section of IIT JEE often tests a student's understanding through problem-solving and application-based questions.
Topics like quadratic inequalities and trigonometric functions are commonly seen because they integrate multiple concepts tested in these exams:

• With quadratic inequalities, students must exhibit proficiency in solving and analyzing these inequalities through factoring, using the quadratic formula, or graphing methods.
• In trigonometry, familiarity with the properties and graphs of functions, including inverse and main functions, is indispensable.

Successfully approaching these problems, as in the given exercise, requires a mix of conceptual understanding and analytical skills. Mastery over these concepts not only helps in tackling the exam effectively but also builds a solid foundation for future studies in engineering and mathematics.