Question

\(\sin \left[\cot ^{-1}\left\\{\cos \left(\tan ^{-1} x\right)\right\\}\right]\) is (a) 1 (b) \(\sqrt{\frac{\mathrm{x}^{2}-1}{\mathrm{x}^{2}+2}}\) (c) \(\sqrt{\frac{x-2}{x^{2}+1}}\) (d) \(\sqrt{\frac{x^{2}+1}{x^{2}+2}}\)

Short answer

Answer: (d) \(\sqrt{\frac{x^{2}+1}{x^{2}+2}}\)

Step-by-step solution

Find the value of \(\tan^{-1}(x)\)

We are given the expression \(\tan^{-1}(x)\). There is no direct simplification available for this expression, so we will leave it as it is.

Find the value of \(\cos(\tan^{-1}(x))\)

Let \(\alpha = \tan^{-1}(x)\). Then, \(\tan(\alpha) = x\). Using the identity \(\tan^2(\alpha) + 1 = \sec^2(\alpha)\) (where \(\sec(\alpha)\) is the reciprocal of \(\cos(\alpha)\)), we have: \[x^{2} + 1 = \frac{1}{\cos^{2}(\alpha)}\] So, we get the value of \(\cos(\alpha)\) (or \(\cos(\tan^{-1}(x))\)) as follows: \[\cos(\tan^{-1}(x)) = \cos(\alpha) = \sqrt{\frac{1}{x^{2}+1}}\]

Find the value of \(\cot^{-1}\{\cos(\tan^{-1}(x))\}\)

Now, we need to find the value of \(\cot^{-1}(\cos(\tan^{-1}(x)))\). Let \(\beta = \cot^{-1}(\cos(\tan^{-1}(x)))\), so \(\cot(\beta)=\cos(\tan^{-1}(x))\). Then we have: \[ \cot(\beta) = \sqrt{\frac{1}{x^{2}+1}} \] Sqaring both sides, we get: \[ \cot^2(\beta) = \frac{1}{x^{2}+1} \] Using identity \(\cot^2(\beta)+1=\csc^2(\beta)\) (where \(\csc(\beta)\) is the reciprocal of \(\sin(\beta)\)), we have: \[ \csc^2(\beta) = x^{2}+1+1 \] So, we can find the value of \(\sin(\beta)\) (or \(\sin(\cot^{-1}(\cos(\tan^{-1}(x))))\)) as follows: \[ \sin(\cot^{-1}(\cos(\tan^{-1}(x)))) = \sin(\beta) = \sqrt{\frac{1}{x^{2}+2}} \]

Compare the expression with given options

At this point, we have found that: \[ \sin(\cot^{-1}(\cos(\tan^{-1}(x)))) = \sqrt{\frac{1}{x^{2}+2}} \] Comparing this expression with the given options, we find that the answer is: (d) \(\sqrt{\frac{x^{2}+1}{x^{2}+2}}\)

Key concepts

Inverse Trigonometric Functions

Inverse trigonometric functions are the opposites of the basic trigonometric functions, allowing us to find the angle when given a ratio. For example, \(\tan^{-1}(x)\) gives us the angle whose tangent is \(x\). This is particularly useful when dealing with expressions or equations where the angle itself is the unknown.

Inverse functions are crucial in solving trigonometric equations because they help convert trigonometric ratios back into angles. These functions have specific domains and ranges, ensuring that they are properly defined. Here's a quick look at the range for some inverse trigonometric functions:

• \(\sin^{-1}(x)\): Range is \([-\frac{\pi}{2}, \frac{\pi}{2}]\)
• \(\cos^{-1}(x)\): Range is \([0, \pi]\)
• \(\tan^{-1}(x)\): Range is \([-\frac{\pi}{2}, \frac{\pi}{2}]\)

The exercise makes use of \(\tan^{-1}(x)\) and \(\cot^{-1}(x)\), which are inverse functions for \(\tan\) and \(\cot\) respectively. Proper understanding of these functions allows us to move from a trigonometric ratio to an allocatable angle, enabling subsequent evaluations in the expression.

Trigonometric Simplification

Trigonometric simplification involves transforming complex trigonometric expressions into simpler forms. This is often accomplished using known identities, such as Pythagorean identities, to make the expression easier to work with.

In the given exercise, simplification becomes critical when evaluating expressions like \(\cos(\tan^{-1}(x))\). We use the identity \(\tan^2(\alpha) + 1 = \sec^2(\alpha)\) to get:

• From \(\tan(\alpha) = x\) and \(\sec^2(\alpha) = x^2 + 1\), we find \(\cos(\alpha) = \sqrt{\frac{1}{x^2 + 1}}\)

By simplifying trigonometric expressions, we reduce the complexity of calculations and make the overall problem more manageable. When these identities are combined logically, they help in unveiling intricate relationships within the expressions allowing steps to progress towards the solution.

Analytic Trigonometry

Analytic trigonometry uses algebraic and geometric principles to solve problems involving trigonometric functions. It often involves arranging and manipulating expressions using known trigonometric identities and formulas.

Consider solving the problem \(\sin(\cot^{-1}(\cos(\tan^{-1}(x))))\), which involves progressing through several layers of trigonometric expressions. Each step uses a piece of analytic trigonometry to simplify and resolve these functions, aiming to match a given solution.

• For instance, converting \(\cos(\tan^{-1}(x))\) to simpler terms through Pythagorean identity.
• Applying inverse trigonometric identities to find specific angle values.

This process highlights the importance of manipulating and working through layers of trigonometric functions and identities to achieve a clearer expression or answer. In this exercise, solving the equation required understanding how to "unwrap" each layer logically, step-by-step, leading to the given solution (d). Understanding these processes is crucial as it forms the foundation of more complex trigonometric analyses.