Question

If \(0

Short answer

The correct answer is (B) \(x\).

Step-by-step solution

Understanding the trigonometric identity

Recognize that for any angle \(\theta\), \(\sin^2\theta + \cos^2\theta = 1\). The expression inside the brackets resembles this identity, suggesting we can simplify it.

Simplify the expression inside the square brackets

Simplify the expression within the square brackets using the identity from Step 1. We have \(\{x\cos(\cot^{-1}x)+\sin(\cot^{-1}x)\}^2 - 1 = (x^2\cos^2(\cot^{-1}x) + 2x\cos(\cot^{-1}x)\sin(\cot^{-1}x) + \sin^2(\cot^{-1}x)) - 1\). Applying the Pythagorean identity gives us \(x^2\cos^2(\cot^{-1}x) + \sin^2(\cot^{-1}x) = x^2(1 - \sin^2(\cot^{-1}x)) + \sin^2(\cot^{-1}x)\).

Apply the definition of cotangent

Use the definition of \(\cot(\theta)\) to express \(\sin(\theta)\) and \(\cos(\theta)\) in terms of \(x\). Since \(\cot(\cot^{-1}x) = x\), we have \(\theta = \cot^{-1}x\), which implies \(\cot(\theta) = \frac{1}{\tan(\theta)} = \frac{\cos(\theta)}{\sin(\theta)} = x\). We can find \(\sin(\theta)\) and \(\cos(\theta)\) using the right-angled triangle with sides \(1\), \(x\), and hypotenuse \(\sqrt{1+x^2}\). Thus, \(\sin(\cot^{-1}x) = \frac{1}{\sqrt{1+x^2}}\) and \(\cos(\cot^{-1}x) = \frac{x}{\sqrt{1+x^2}}\).

Substitute the trigonometric values back into the simplified expression

Replace \(\sin(\cot^{-1}x)\) and \(\cos(\cot^{-1}x)\) in the expression with their equivalents found in Step 3: \(x^2(1 - (\frac{1}{\sqrt{1+x^2}})^2) + (\frac{1}{\sqrt{1+x^2}})^2 = x^2(1 - \frac{1}{1+x^2}) + \frac{1}{1+x^2} = x^2\frac{x^2}{1+x^2} + \frac{1}{1+x^2} = \frac{x^4 + 1}{1+x^2}\). After simplifying, we get \(\frac{x^2}{1+x^2}\).

Apply the square root to the simplified expression

Take the square root of the expression obtained in Step 4, which gives us \(\sqrt{\frac{x^2}{1+x^2}} = \frac{|x|}{\sqrt{1+x^2}}\). Since \(0<x<1\), \(x\) is positive, and we can remove the absolute value to get \(\frac{x}{\sqrt{1+x^2}}\).

Multiply by \(\sqrt{1+x^2}\) outside the square root

Multiply the expression \(\frac{x}{\sqrt{1+x^2}}\) by \(\sqrt{1+x^2}\) from outside the square root to get the final solution: \(\sqrt{1+x^2} \times \frac{x}{\sqrt{1+x^2}} = x\).

Key concepts

Cotangent

The cotangent function is one of the six fundamental trigonometric functions, and understanding it is key to solving many trigonometry problems. In the context of a right-angled triangle, cotangent is defined as the ratio of the adjacent side to the opposite side. Mathematically, it's represented as \( \cot(\theta) = \frac{\mathrm{adjacent}}{\mathrm{opposite}} = \frac{1}{\tan(\theta)} \).

When dealing with inverse trigonometric functions, such as \( \cot^{-1} \), we're essentially finding the angle whose cotangent is the given number. In the provided example, \( \cot^{-1}x \) is the angle whose cotangent is \( x \). Solving such expressions requires you to visualize or construct a right-angled triangle that embodies the ratio presented by the cotangent value. This enables you to determine the other trigonometric functions of the angle in question, as was done in the solution steps.

Pythagorean Identity

The Pythagorean identity is a critical principle in trigonometry that stems from the Pythagorean theorem. This identity links the sine and cosine of an angle using the equation \( \sin^2(\theta) + \cos^2(\theta) = 1 \).

Application in the Exercise

In our exercise, the Pythagorean identity helps simplify the expression within the square brackets. The identity demonstrates that the sum of the squares of sine and cosine values of any angle will always equal one, irrespective of the angle's measure. In practical terms, this allows you to transform and simplify expressions involving sine and cosine, providing a path to solve the equation, as seen in the step-by-step solution. This identity is a fundamental tool for understanding and manipulating trigonometric expressions.

Inverse Trigonometric Functions

Inverse trigonometric functions are the inverses of the trigonometric functions, serving to determine the angles when the function values are known. These include inverse sine (\(\sin^{-1}\)), inverse cosine (\(\cos^{-1}\)), and inverse cotangent (\(\cot^{-1}\)) among others. They are crucial for solving equations where the goal is to find the measure of an angle given a trigonometric ratio.

In the original problem, we encounter \(\cot^{-1}x\), which means we are looking for an angle whose cotangent equals \(x\). To solve this, we use a right triangle approach and express the other trigonometric functions based on the known ratio. As was shown in the solution, \(\cot^{-1}x\) provides a way to concretely find specific values of sine and cosine, which then makes it possible to simplify and solve the rest of the expression.