Problem 4 - Comprehensive Trigonometry for IIT JEE Main and Advanced Rejaul Makshud MeGraw Hill

Question

Prove that \(\sin \left(\cot ^{-1}\left(\tan \left(\cos ^{-1} x\right)\right)\right)=x \quad \forall x \in(0,1]\)

Step-by-Step Solution

Answer

Indeed, it's been confirmed that \(\sin \left(\cot ^{-1}\left(\tan \left(\cos ^{-1}x\right)\right)\right) = x\) for all \(x \in(0,1]\). This is a result of accurate utilization of the relationships between trigonometric functions and their inverses.

Step 1: Decompose the Innermost Functions

Begin with the innermost function, which is \(\cos^{-1}x\). Since the cosine function is the ratio of the adjacent side to the hypotenuse in a right triangle, you can imagine this as the angle \(\theta\) in a right triangle such that \(\cos\theta = x\). Therefore, the adjacent side is x and the hypotenuse is 1. Using the Pythagorean theorem, the opposite side is \(\sqrt{1 - x^2}\). Hence, \(\tan(\cos^{-1}x) = \frac{\sqrt{1 - x^2}}{x}\).

Step 2: Apply Inverse Cotangent

Next, you can simplify \(\cot^{-1}\left(\tan \left(\cos ^{-1}x\right)\right)\) as \(\cot^{-1}\left(\frac{x}{\sqrt{1 - x^2}}\right)\). The cotangent function is just the reciprocal of the tangent function, implying that this is an angle \(\phi\) for which \(\tan\phi = \frac{x}{\sqrt{1 - x^2}}\). Thus, you can imagine \(\phi\) as an angle in a right triangle such that the opposite side is x and the adjacent side is \(\sqrt{1 - x^2}\). By the Pythagorean theorem, the hypotenuse in this case is 1.

Step 3: Apply Sine Function

Finally, \(\sin \left(\cot ^{-1}\left(\tan \left(\cos ^{-1}x\right)\right)\right)\) is simplified as \(\sin(\phi) = \frac{x}{1} = x\). So, it's proven that \(\sin \left(\cot ^{-1}\left(\tan \left(\cos ^{-1} x\right)\right)\right) = x\).

Key Concepts

Inverse Trigonometric Functions

Understanding inverse trigonometric functions can be a bit complex, but they're quite fascinating! These functions basically help us find angles when we have a known trigonometric value. For example, if we know \( \cos \theta = x \), the inverse cosine function, \( \cos^{-1} x \), gives us the angle \( \theta \) for which this is true.
You can think of inverse trigonometric functions in terms of their practical use:

• \( \cos^{-1} x \): Finds the angle whose cosine is \( x \).
• \( \tan^{-1} x \): Finds the angle whose tangent is \( x \).
• \( \cot^{-1} x \): Finds the angle whose cotangent is \( x \).

In our problem, the use of these functions helps deconstruct a seemingly complicated expression into understandable parts, eventually simplifying it to \( x \).
This highlights their power and usefulness in solving trigonometric equations.

Right Triangle Trigonometry

Right triangle trigonometry is a fundamental concept that connects angles to side lengths in a triangle. Each right triangle features one angle of 90 degrees, and its properties let us use trigonometric ratios easily.
Consider the following:

• Sine (sin): Ratio of the opposite side to the hypotenuse.
• Cosine (cos): Ratio of the adjacent side to the hypotenuse.
• Tangent (tan): Ratio of the opposite side to the adjacent side.

In our exercise, imagining \( \theta \) using \( \cos^{-1} x \) and \( \phi \) using \( \cot^{-1} \), leverages these ratios to break down calculations.
We build up each part of the expression using these trigonometric identities, visualizing the relationships between the sides of a right triangle.

Pythagorean Theorem

The Pythagorean Theorem is a key mathematical relationship in right triangles, expressed as \( a^2 + b^2 = c^2 \), where \( c \) is the hypotenuse, and \( a \) and \( b \) are the other two sides. This theorem allows us to calculate a missing side of the triangle when we know the other two.
In our problem, using the theorem allows us to find the opposite side once we know the adjacent side (\( x \)) and hypotenuse (\( 1 \)).
Here's how we use it:

• Find the opposite side: \( \sqrt{1 - x^2} \).
• Calculate tangent: \( \frac{\sqrt{1 - x^2}}{x} \).

This process illustrates how geometry and algebra nicely complement each other, enabling us to prove the given trigonometric identity. It’s a clear demonstration of logical problem-solving using essential mathematical tools.