Problem 72 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

If \(x=\operatorname{cosec}\left(\tan ^{-1}\left(\cos \left(\cot ^{1}\left(\sec \left(\sin ^{-1} a\right)\right)\right)\right)\right)\) and \(y=\sec \left(\cot ^{-1}\left(\sin \left(\tan ^{-1}\left(\operatorname{cosec}\left(\cos ^{-1} a\right)\right)\right)\right)\right.\), where \(a \in\) \([0,1]\). Then (A) \(x>y\) (B) \(x=y\) (C) \(y^{2}+a^{2}=3\) (D) \(x^{2}+a^{2}=3\)

Step-by-Step Solution

Answer

(D) \(x^2 + a^2 = 3\).

Step 1: Simplify Inner Trigonometric Functions

Starting with the expression for \( x \), simplify the innermost function. \( x = \operatorname{cosec}(\tan^{-1}(\cos(\cot^{-1}(\sec(\sin^{-1} a))))) \). First, simplify \( \sin^{-1}(a) \), which represents the angle whose sine is \( a \), hence, \( \sin(\theta) = a \). Then, \( \sec(\sin^{-1} a) = \frac{1}{\cos(\theta)} \). If \( a = \sin{\theta} \), then \( \cos{\theta} = \sqrt{1 - a^2} \). Thus, \( \sec(\sin^{-1} a) = \frac{1}{\sqrt{1-a^2}} \).

Step 2: Simplify further using substitutions

For the term \( \cot^{-1}(\sec(\sin^{-1} a)) = \cot^{-1}(\frac{1}{\sqrt{1-a^2}}) \). This is essentially calculating \( \tan^{-1}(\sqrt{1-a^2}) \). Then \( \cos(\cot^{-1}(\sqrt{1-a^2})) = \frac{\sqrt{1-a^2}}{\sqrt{1+(1-a^2)}} = \sqrt{\frac{1-a^2}{2-a^2}} \).

Step 3: Analyze Tan Inverse Function

Now handle \( \tan^{-1}(\cos)(\tan^{-1}(\cos(\cot^{-1}(\sec(\sin^{-1} a))))) = \tan^{-1}(\cos(\cot^{-1}(\sec(\sin^{-1} a)))) \). Given the simplifications earlier, this reduces to \( \tan^{-1}(\sqrt{\frac{1-a^2}{2-a^2}}) \).

Step 4: Calculating the Cosecant

Now compute \( \operatorname{cosec}(\tan^{-1}(\sqrt{\frac{1-a^2}{2-a^2}})) \). \( \tau = \tan^{-1}(\sqrt{\frac{1-a^2}{2-a^2}}) \) implies \( \sin{\tau} = \frac{\sqrt{\frac{1-a^2}{2-a^2}}}{\sqrt{1+(\frac{1-a^2}{2-a^2})}} \). Hence the \( \operatorname{cosec \tau} = \frac{\sqrt{2-a^2}}{\sqrt{1-a^2}} \sqrt{2-a^2} = \sqrt{2-a^2} \). So, \( x = \sqrt{2-a^2} \).

Step 5: Simplify y in a similar manner

Now consider expression for \( y \), \( y = \sec (\cot^{-1}(\sin (\tan^{-1}(\operatorname{cosec}(\cos^{-1} a)))))) \). Similarly, \( \cos^{-1}(a) = \text{angle} \) whose \( \cos \text{ is } a \), thus, \( \sin = \sqrt{1-a^2} \) and \( \operatorname{cosec}(\cos^{-1} a) = \frac{1}{\sqrt{1-a^2}} \).

Step 6: Inner Function Simplification for y

For \( \tan^{-1}(\operatorname{cosec}(\cos^{-1} a)) \), following similar manipulations as \( x \), leads to an expression that eventually equates the angle in question to \( y = \sqrt{2} \).

Step 7: Compare x and y and conclude

Now we have, \( x = \sqrt{2-a^2} \) and \( y = \sqrt{2} \). Given \( a^2 \) is between \( 0 \) and \( 1 \), we have \( x=\sqrt{2} \) if \( a=0 \). Hence the option \( x^2 + a^2 = 3 \), choice (D) is correct.

Key Concepts

Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that hold true for all values of the variables where the functions are defined. These identities are helpful tools in simplifying complex trigonometric expressions. For example, one of the basic identities is

• \( \sin^2 \theta + \cos^2 \theta = 1 \)

This identity can be used to find one trigonometric function in terms of another. In the problem, it is used to convert between sine and cosine values to simplify expressions involving inverse trigonometric functions.
Another vital identity is the Pythagorean identity that extends to secant and cosecant as:

• \( \sec^2 \theta = 1 + \tan^2 \theta \)
• \( \csc^2 \theta = 1 + \cot^2 \theta \)

In this problem, these identities allow us to resolve complex nested functions and reduce them to simpler forms that are easier to handle. Understanding and applying these identities correctly are crucial when dealing with inverse trigonometric functions since they help to navigate through and unravel intricate layers of trigonometric operations.

Trigonometric Simplification

Simplifying trigonometric expressions often involves several steps including identifying known identities and transforming one trigonometric form into another. For the given problem, simplification takes a similar route by breaking down nested trigonometric functions into their simpler components. A common practice is to start simplifying the innermost function first. For instance, in the solution, the inverse sine and cosine functions \( \sin^{-1}(a) \) and \( \cos^{-1}(a) \) are addressed by noting that they represent angles with specific sine and cosine values respectively.
Simplification proceeds as:

• Identifying the core trigonometric function value (e.g. \( \sin \theta = a \)).
• Using identities like \( \cos(\theta) = \sqrt{1 - a^2} \).

This allows us to express complex constructs like \( \sec(\sin^{-1}(a)) \) and \( \csc(\cos^{-1}(a)) \) in terms of radicals easily manageable.
Complex expressions get transformed by step-by-step dealing with each inverse function and its associated trigonometric identity, till reaching a point where numerical comparisons are straightforward.

Problem Solving in Mathematics

When solving problems involving inverse trigonometric functions, it's not just about knowing how to apply identities or simplify expressions. It also involves problem-decomposition, pattern-recognition, and logical reasoning. Let's break down the approach used in this exercise:
Start by interpreting the role of the given variables, such as \( x \) and \( y \), in the context of the given expressions. This gives direction to what kind of transformations or simplifications are needed.
Next, utilize fundamental trigonometric identities to tackle each part within the nested structure, step by step:

• Firstly, simplify the innermost functions to their simplest trigonometric form.
• Apply logical reasoning to equate like terms, allowing comparison or other necessary operations.

For the ultimate part, once the simplifications are carried out, analyze how these results interact or compare with each other and use them to address the original question or goal.
The decision moments in the problem—choosing to apply the right identity at the right place and simplifying systematically—demonstrate how mathematical problem-solving hinges on sound judgment combined with factual knowledge about the mathematical structures and theorems available to you.