Problem 39 - Mathematics for IIT JEE Main and Advanced Differential Calculus Algebra Trigonometry

Question

$$ \sin (\pi+\theta) \sin (\pi-\theta) \operatorname{cosec}^{2} \theta=-1 $$

Step-by-Step Solution

Answer

The given equation \(\sin(\pi+\theta) \sin(\pi-\theta) \operatorname{cosec}^{2}\theta = -1\) can be proven true for any angle θ, where \(\sin(\theta) \neq 1\) and \(\sin(\theta) \neq -1\), by applying trigonometric identities and simplifying the equation as follows: \((- \sin(\theta))\sin(\theta)\left(\frac{1}{\sin^2(\theta)}\right)^2 = -1\), which leads to \(\operatorname{cosec}(\theta) = \frac{1}{\sin(\theta)} = -1\).

Step 1: Identify the trigonometric identities to be used

We will use the following identities: 1. \(\sin(\pi + \theta) = -\sin(\theta)\) 2. \(\sin(\pi - \theta) = \sin(\theta)\) 3. \(\operatorname{cosec}(\theta) = \frac{1}{\sin(\theta)}\) Step 2: Apply the trigonometric identities

Step 2: Rewrite the given equation using the identified identities

Substitute the trigonometric identities into the given equation: \((- \sin(\theta))\sin(\theta)\left(\frac{1}{\sin^2(\theta)}\right)^2 = -1\) Step 3: Simplify the equation

Step 3: Combine and simplify the terms

Rewrite the equation as: \(\sin^2(\theta)\left(\frac{1}{\sin^2(\theta)}\right)^2 = 1\) Now, cancel out \(\sin^2(\theta)\) terms: \(\left(\frac{1}{\sin^2(\theta)}\right)^2 = 1\) Step 4: Prove the relation equals -1

Step 4: Square root both sides and simplify further

Take the square root of both sides and simplify: \(\frac{1}{\sin(\theta)} = \pm1\) It's important to note that \(\sin(\theta) \neq 1\) and \(\sin(\theta) \neq -1\), so: \(\operatorname{cosec}(\theta) = \frac{1}{\sin(\theta)} = -1\) Step 5: Conclusion

Step 5: State the conclusion

The given equation \(\sin (\pi+\theta) \sin (\pi-\theta) \operatorname{cosec}^{2} \theta = -1\) is proven to be true for any angle θ, where \(\sin(\theta) \neq 1\) and \(\sin(\theta) \neq -1\).

Key Concepts

Trigonometric Identities

Trigonometric identities are equalities involving trigonometric functions that hold for all values of the involved variables. These identities are useful in simplifying complex trigonometric expressions and solving trigonometric equations. A basic understanding of these identities is essential in mastering trigonometry. To prove trigonometric identities, like in our exercise, we often start by recognizing patterns and using known identities such as the angle sum and difference identities. For example, \(\sin(\pi + \theta) = - \sin(\theta)\) and \(\sin(\pi - \theta) = \sin(\theta)\) demonstrate how adding or subtracting \(\pi\) to an angle \(\theta\) can change the sign of the sine function. These are based on the unit circle and the symmetries of sine and cosine functions.

In the given problem, we needed to use these identities to transform and simplify the equation. Roughly speaking, these identities are the tools that allow us to manipulate and eventually simplify trigonometric expressions to something more manageable or to prove that a particular relation holds true for all angles, leading us to the final proof of the identity.

Cosecant Function

The cosecant function, denoted as \(\operatorname{cosec}(\theta)\) or \(\csc(\theta)\), is the reciprocal of the sine function. It is defined as \(\operatorname{cosec}(\theta) = \frac{1}{\sin(\theta)}\), where \(\theta\) is an angle measured in radians, and \(\sin(\theta)\) cannot equal zero (since division by zero is undefined). The function has an undefined value at \(\theta = 0\) and \(\theta = \pi\) among others, as these are the angles where \(\sin(\theta) = 0\). It is crucial to recognize the behavior of the cosecant function when proving trigonometric identities involving \(\cosec\), as it can lead to simplifications or specific values under certain conditions.

In our problem, we needed to square the cosecant function, leading to a dramatic simplification when taking into account the reciprocal nature of the cosecant and sine functions. This insight is pivotal for proving the given identity and is a typical step when dealing with reciprocal trigonometric functions.

Simplifying Trigonometric Expressions

Simplifying trigonometric expressions is a common task in trigonometry, involving several techniques to rewrite expressions into a more manageable form. This can consist of using basic algebraic operations, trigonometric identities, and properties of trigonometric functions. The end goal is to either prove an identity, solve an equation, or calculate a function's exact value.

In the context of our problem, simplification involved applying known identities, canceling out identical terms, and recognizing that \(\sin(\theta)\) essentially 'cancels out' when it appears in both the numerator and denominator of a fraction. The process requires careful manipulation of the terms and a good understanding of the functions involved. Crucial steps in simplification include combining like terms, factoring, and rationalizing denominators. These steps help to reveal the underlying simplicity of an expression that might initially seem complex, much like the process we used to prove the trigonometric identity at hand.