Question

Evaluate: \(\operatorname{Lt}_{\theta \rightarrow 0} \cot \theta-\operatorname{cosec} \theta\) (1) 0 (2) 2 (3) 4 (4) None of these

Short answer

Answer: None of these

Step-by-step solution

Rewrite the expression using basic trigonometric functions

Recall that \(\cot \theta = \frac{1}{\tan \theta}\) and \(\operatorname{cosec} \theta = \frac{1}{\sin \theta}\). Thus, the given expression becomes: $$\cot \theta - \operatorname{cosec} \theta = \frac{1}{\tan \theta} - \frac{1}{\sin \theta}$$

Write the expression in a common denominator

To be able to combine the fractions, we need a common denominator. In this case, it's \(\sin\theta \tan\theta\): $$\frac{1}{\tan \theta} - \frac{1}{\sin \theta} = \frac{\sin\theta -\tan\theta}{\sin\theta \tan\theta}$$

Simplify the numerator

Replace the \(\tan\theta\) in the numerator with its equivalent expression \(\frac{\sin\theta}{\cos\theta}\): $$\frac{\sin\theta -\tan\theta}{\sin\theta \tan\theta} = \frac{\sin\theta - \frac{\sin\theta}{\cos\theta}}{\sin\theta \tan\theta} = \frac{\sin\theta\cos\theta - \sin\theta}{\sin\theta\cos\theta\tan\theta}$$

Simplify the expression

We can factor out \(\sin\theta\) from the numerator and cancel it with the \(\sin\theta\) term in the denominator: $$\frac{\sin\theta\cos\theta - \sin\theta}{\sin\theta\cos\theta\tan\theta} = \frac{\sin\theta(\cos\theta - 1)}{\sin\theta\cos\theta\tan\theta} = \frac{\cos\theta - 1}{\cos\theta\tan\theta}$$

Evaluate the limit

Now that we have simplified the expression, we can evaluate the limit as \(\theta\) approaches \(0\): $$\operatorname{Lt}_{\theta \rightarrow 0} \frac{\cos\theta - 1}{\cos\theta\tan\theta}$$ As we approach 0, \(\cos(0) = 1\), and \(\tan(0) = 0\). Thus, the limit becomes: $$\frac{1 - 1}{1\times0}$$ This cannot be evaluated directly, as it's of the indeterminate form `\(\frac{0}{0}\)`. Therefore, the answer is (4) None of these.

Key concepts

Trigonometric Identities

Understanding trigonometric identities is essential for evaluating limits that involve trigonometric functions. An identity in trigonometry is an equation that is true for all angle measurements in its domain. Key identities include the Pythagorean identities, quotient identities, and reciprocal identities.

For example, reciprocal identities allow us to transform \(\cot \theta\) into \(\frac{1}{\tan \theta}\) and \(\operatorname{cosec} \theta\) into \(\frac{1}{\sin \theta}\). These transformations are the starting point as they reframe trigonometric functions in terms that make it easier to find a common denominator and combine terms.

Another useful set of identities involves the quotient identities, such as \(\tan \theta = \frac{\sin \theta}{\cos \theta}\), which comes into play when simplifying the numerator in the problem's expression. Understanding these relationships can greatly simplify the limit evaluation process, especially when dealing with complex expressions.

Common Denominators in Algebra

When combining fractions, finding a common denominator is a crucial step. In algebra, a common denominator refers to a common multiple of the denominators of two or more fractions. This allows the fractions to be combined into a single expression, which can then be simplified or manipulated further.

In our exercise, seeking a common denominator enables us to combine \(\frac{1}{\tan \theta}\) and \(\frac{1}{\sin \theta}\). The common denominator, \(\sin \theta \tan \theta\), is found by multiplying the original denominators together. It is the simplest mathematical structure that can host both denominators. By putting the trigonometric functions over this common denominator, we create a single fraction, which can then be simplified further. This process requires a fundamental understanding of algebra and is widely used across different areas of mathematics, highlighting its foundational importance.

Indeterminate Forms in Calculus

Indeterminate forms occur in calculus when evaluating a limit leads to an expression that is not immediately clear or is undefined, such as \(\frac{0}{0}\), \(\frac{\infty}{\infty}\), \(0 \times \infty\), among others. In the given problem, after applying trigonometric identities and finding a common denominator, we are left with such a form when \(\theta \rightarrow 0\).

To resolve indeterminate forms, calculus provides various techniques such as L'Hôpital's rule, algebraic manipulation, or Taylor series expansion. While these methods are beyond the scope of this exercise, it's crucial to recognize indeterminate forms as signals that further work is needed to evaluate a limit. In this case, our expression simplifies to \(\frac{0}{0}\) as \(\theta\) approaches zero, which means we cannot determine the limit without additional techniques, resulting in the correct answer being 'None of these' (option 4). This teaches us the importance of indeterminate forms in identifying when a straightforward limit calculation is insufficient for finding an answer.