Problem 53 - A Complete Resource Book in Mathematics for JEE Main

Question

If \(y=\sin ^{2} \theta+\operatorname{cosec}^{2} \theta, \theta \neq 0\) then: (A) \(y=0\) (B) \(y \leq 2\) (C) \(y \geq-2\) (D) \(y \geq 2\)

Step-by-Step Solution

Answer

The correct answer is (D) \(y \geq 2\).

Step 1: Recall Fundamental Trigonometric Identites

First, recall the identity for cosecant: \( \csc \theta = \frac{1}{\sin \theta} \). Therefore, \( \csc^2 \theta = \left( \frac{1}{\sin \theta} \right)^2 = \frac{1}{\sin^2 \theta} \).

Step 2: Substitute Known Identities

Replace \( \csc^2 \theta \) in the original equation with the identity: \[y = \sin^2 \theta + \frac{1}{\sin^2 \theta}\]

Step 3: Let \( x = \sin^2 \theta \)

To simplify, let \( x = \sin^2 \theta \). Then the equation becomes:\[y = x + \frac{1}{x}\].

Step 4: Determine Minimal Value of the Expression

Recall AM-GM (Arithmetic Mean - Geometric Mean) inequality: For two positive numbers \( a \) and \( b \), we have \( \frac{a+b}{2} \geq \sqrt{ab} \). Applying it to \( x \) and \( \frac{1}{x} \):\[\frac{x + \frac{1}{x}}{2} \geq \sqrt{x \cdot \frac{1}{x}} = 1\]Therefore, \( x + \frac{1}{x} \geq 2 \).

Step 5: Conclude the Minimal Value

From the AM-GM inequality, we conclude that the minimum value of \( x + \frac{1}{x} \) is \( 2 \). Thus, \( y \geq 2 \).

Step 6: Final Step: Choose the Correct Option

Based on the analysis, the inequality \( y \geq 2 \) holds. This corresponds to option (D).

Key Concepts

Trigonometric Identities

Understanding trigonometric identities is crucial when tackling problems involving angles and trigonometric functions such as sine, cosine, and cosecant. A trigonometric identity is a true equation involving trigonometric functions for all values within a domain where the functions are defined. For instance, the identity related to cosecant is:\[\csc \theta = \frac{1}{\sin \theta}\]This identity means that the cosecant of an angle \(\theta\) is the reciprocal of the sine of \(\theta\). When we square both sides, we obtain:\[\csc^2 \theta = \left( \frac{1}{\sin \theta} \right)^2 = \frac{1}{\sin^2 \theta}\]This is very handy for transforming equations involving cosecant into a more familiar form, because it allows substitution with terms of sine, simplifying the equation. In problems like the given exercise, using these identities allows us to express the equation in a form that can be tackled with other mathematical concepts, like inequalities.

AM-GM Inequality

The AM-GM inequality is a fundamental concept in mathematics that connects arithmetic mean and geometric mean of a set of non-negative numbers. It states that for any non-negative real numbers, the arithmetic mean is always greater than or equal to the geometric mean. For two positive numbers \(a\) and \(b\), the inequality is stated as:\[\frac{a + b}{2} \geq \sqrt{ab}\]In the context of the exercise, applying the AM-GM inequality to \(x\) and \(\frac{1}{x}\) leads to:\[\frac{x + \frac{1}{x}}{2} \geq \sqrt{x \cdot \frac{1}{x}} = 1\]Thus, we can conclude that:\[x + \frac{1}{x} \geq 2\]This result tells us that the expression involving the sine function and its reciprocal has a minimum value of 2. The relevance of AM-GM inequality here is strategic; it allows us to determine the lower bound of the expression, which is critical in proving the given inequality \(y \geq 2\). By showing the minimum is 2, it helps identify the correct option in the multiple-choice answer.

Cosecant Function

The cosecant function is one of the reciprocal trigonometric functions, specifically the reciprocal of the sine function. Defined as \(\csc \theta = \frac{1}{\sin \theta}\), it is important to understand the behavior and properties of this function for solving problems like the one presented in the exercise.Key properties of the cosecant function include:

• The function is undefined where the sine is zero, which occurs at integer multiples of \(\pi\).
• It has vertical asymptotes at these points, making it crucial to remember this when solving trigonometric inequalities involving cosecant.
• Because it is the reciprocal of sine, cosecant is always greater than or equal to one, or less than or equal to negative one. This highlights its bounded nature compared to sine.

In the exercise, knowing that \(\csc^2 \theta\) can be expressed as \(\frac{1}{\sin^2 \theta}\) was critical. It allowed the conversion of the expression into a format more conducive to applying the AM-GM inequality, which in turn led to identifying the minimum value of \(y\). By transforming the trigonometric function, the problem is simplified into a basic inequality problem, thus demonstrating the utility of understanding and applying the properties and identities of the cosecant function.