\(A=\tan ^{-1} x+\sec ^{-1} x\), then \(A\) lies in the interval set (a) \([-\infty,-1] \cup[1, \infty]\) (b) \([-\infty,-1] \cup[1, \infty]\) (c) \([-\infty, 1] \cup[1, \infty]\) (d) none of these

When studying trigonometry, you'll come across the function \(\tan^{-1}(x)\), also known as the inverse tangent or arctangent. The purpose of this function is to determine the angle whose tangent is \(x\). Unlike the regular tangent function, which has an infinite range, the range of \(\tan^{-1}(x)\) is restricted to lie between \(-\frac{\pi}{2}\) and \(\frac{\pi}{2}\), which means it can only output angles in this interval.

It's essential to understand that no matter the value of \(x\), whether it's a large positive number, a negative number, or zero, the output of \(\tan^{-1}(x)\) will always fall within the open interval \(\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)\). This ensures that the function is both well-defined and single-valued, which is critical for reliable mathematical problem-solving.

Another important inverse function in trigonometry is \(\sec^{-1}(x)\), known as the inverse secant. The secant function is the reciprocal of the cosine function, and its inverse is used to find an angle whose secant is \(x\). However, since secant has singularities at certain intervals where cosine is zero, the range of \(\sec^{-1}(x)\) is restricted to ensure the function remains single-valued.

The range of \(\sec^{-1}(x)\) is \([0, \pi]\) excluding the point \(\frac{\pi}{2}\), because the secant function is not defined there. So we use interval notation to describe the range as \([0, \frac{\pi}{2}) \cup (\frac{\pi}{2}, \pi]\). It's crucial to remember that \(\sec^{-1}(x)\) only outputs angles within this domain, regardless of the argument \(x\), as long as \(x\) is equal to or greater than 1, or less than or equal to -1.

Inverse trigonometric functions have unique properties that differentiate them from their respective trigonometric functions. One fundamental property is the limited range of these functions, which helps prevent the occurrence of multiple angles corresponding to a single value of \(x\).

Furthermore, the inverse trigonometric functions are continuous and differentiable within their domains, which is important for calculus applications. Additionally, they obey the property that for a given function, say \(\tan^{-1}(x)\), applying the tangent function would return the original value, symbolically presented as \(\tan(\tan^{-1}(x)) = x\), for x within the domain of \(\tan^{-1}(x)\).

These properties are critical when solving equations involving inverse trigonometric functions and understanding their behavior when graphing or analyzing functions within a certain interval.

Interval notation is a mathematical shorthand used to express a range of values in which a variable can exist. It's especially handy when describing domains, ranges, and solutions to inequalities. An interval can either be closed, open, or a combination of both.

A closed interval, indicated by brackets \([\text{ and }]\), includes the endpoints of the interval. For example, \([1, 3]\) means all numbers from 1 to 3, including 1 and 3 themselves. An open interval, indicated by parentheses \((\text{ and }))\), does not include the endpoints. For example, \((1, 3)\) includes numbers greater than 1 and less than 3, but not 1 or 3.

When working with trigonometric functions, you'll often use interval notation to describe the range or domain of these functions, which is essential for correctly understanding and solving problems in trigonometry. Recognizing the meaning of the different brackets and the values they encompass or exclude allows for precise and unambiguous mathematical communication.