Problem 46 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

The value of \(x\) for which \(\sin \left(\cot ^{-1}(1+x)\right)=\cos \left(\tan ^{-1} x\right)\) is (A) \(\frac{1}{2}\) (B) 1 (C) 0 (D) \(-\frac{1}{2}\)

Step-by-Step Solution

Answer

The value of \(x\) is 0, corresponding to option (C).

Step 1: Understanding the problem

We are asked to find the value of \(x\) such that \(\sin(\cot^{-1}(1+x)) = \cos(\tan^{-1}(x))\). This involves inverse trigonometric functions, and we need to simplify and compare both sides to find \(x\).

Step 2: Simplifying the left side

We know that \(\sin(A) = \cos(\frac{\pi}{2} - A)\). Therefore, \(\sin(\cot^{-1}(1+x)) = \cos(\frac{\pi}{2} - \cot^{-1}(1+x))\). As \(\cot^{-1}(y) = \tan^{-1}(\frac{1}{y})\), convert: \(\cot^{-1}(1+x) = \tan^{-1}(\frac{1}{1+x})\).

Step 3: Simplifying the right side

We have \(\cos(\tan^{-1}(x))\) and know the identity \(\cos(\tan^{-1}(x)) = \frac{1}{\sqrt{1+x^2}}\).

Step 4: Comparing both sides

Since both must equal, we now set\(\cos(\tan^{-1}(\frac{1}{1+x})) = \frac{1}{\sqrt{1+x^2}}\). This implies \(\frac{1}{\sqrt{1+(\frac{1}{1+x})^2}} = \frac{1}{\sqrt{1+x^2}}\).

Step 5: Solving for x

Squaring both sides to eliminate the square roots: \(1 + \left(\frac{1}{1+x}\right)^2 = 1 + x^2\). Simplifying: \((1+x)^2 + 1 = (1+x^2)(1+x)^2\). Eventually, solving gives \(x = 0\).

Key Concepts

Trigonometric Identities

Trigonometric identities are fundamental tools used in simplifying and solving equations involving trigonometric functions. They allow us to transform a complex expression into a more manageable form.

For instance, in the original problem \[\sin(\cot^{-1}(1+x)) = \cos(\tan^{-1}(x))\]we used the identity \[\sin(A) = \cos\left(\frac{\pi}{2} - A\right)\] to simplify the left side of the equation. This approach lets us switch a sine function into a cosine function, facilitating comparison with the right side of our equation.

Trigonometric identities, like \[\tan^{-1}(y) = \cot^{-1}\left(\frac{1}{y}\right)\]are crucial to these transformations. Using identities is akin to recognizing patterns in algebra.

• These identities are the backbone of manipulating angles and expressions.
• Learning them helps greatly in breaking down seemingly complex trigonometric equations.

Simplification of Equations

Simplifying trigonometric equations often involves transforming both sides of an equation to match or reveal a common term. This makes the process of finding solutions more straightforward.

In the exercise, simplification involved using identities to turn complex inverse trigonometric functions into simpler forms.

Starting with the left side:

• We convert \( \sin(\cot^{-1}(1+x)) \) to a cosine function by using trigonometric identities, then this function becomes easier to analyze.

Similarly, the right side \[\cos(\tan^{-1}(x)) = \frac{1}{\sqrt{1+x^2}}\]was already simplified using a known identity.

The goal in simplification is to reduce the equation to its purest form. This not only makes it easier to solve but also to compare or relate the two sides of an equation. Remember:

• Identify and apply the right identities or properties.
• Balance both sides for easier solving.

Solving Trigonometric Equations

After simplifying a trigonometric equation, solving it becomes a logical progression of comparing, balancing, and sometimes using algebraic manipulation.

In our problem, we simplified both sides and ended up with:

• This facilitated the comparison by eliminating square roots: \( \frac{1}{\sqrt{1+(\frac{1}{1+x})^2}} = \frac{1}{\sqrt{1+x^2}} \)

Then, the next step was to square both sides, resulting in:

• \(1 + \left(\frac{1}{1+x}\right)^2 = 1 + x^2 \)

Solving required equating and manipulating these expressions algebraically. This typically follows:

• Remove unwanted fractions by multiplying through or simplifying fractions with a common denominator.
• Factor where possible or apply other algebraic strategies to find potential values for the variable.

Finally, solving gives the solution \(x = 0\). Solving trigonometric equations demands careful substitution, patient fact-checking, and precise operations.