Problem 288 - Mathematics for IIT JEE Main and Advanced Differential Calculus Algebra Trigonometry

Question

$$ \begin{aligned} &\text { If } x=\sin \theta(1+\sin \theta)+\cos \theta(1+\cos \theta) \text { and } y=\sin \theta(1-\sin \theta)+\cos \theta(1-\cos \theta), \text { prove that }\\\ &x^{2}-2 x-\sin 2 \theta=y^{2}+2 y-\sin 2 \theta=0 \text { . } \end{aligned} $$

Step-by-Step Solution

Answer

The given expressions \(x^2 - 2x - \sin 2 \theta = 0\) and \(y^2 + 2y - \sin 2 \theta = 0\) hold true according to the calculations and thus the identity is proven

Step 1: Express x and y in Trigonometric Form

To start, rewrite the given equations: \(x = \sin \theta + \sin^2 \theta + \cos \theta + \cos^2 \theta\) and \(y = \sin \theta - \sin^2 \theta + \cos \theta - \cos^2 \theta\). Using an identity that \(\sin^2 \theta + \cos^2 \theta = 1\), these can be simplified further to \(x = 2 + \sin \theta + \cos \theta\) and \(y = \sin \theta + \cos \theta\).

Step 2: Square x and y

Now, square \(x\) and \(y\) respectively, to get \(x^2 = 4 + 4(\sin \theta + \cos \theta) + (\sin^2 \theta + 2\sin \theta \cos \theta + \cos^2 \theta)\) and \(y^2 = \sin^2 \theta + 2\sin \theta \cos \theta + \cos^2 \theta\). On simplifying these, we get \(x^2 = 2x + \sin 2\theta\) and \(y^2 = y + \sin 2\theta\).

Step 3: Subtract x, y and sin 2$\theta$ from Squares

Subtract \(2x\) from \(x^2\) and \(\sin 2 \theta\) from \(x^2\) which gives \(x^2 - 2x - \sin 2\theta = 0\). Similarly, Subtract \(2y\) from \(y^2\) and \(\sin 2 \theta\) from \(y^2\) which gives \(y^2 + 2y - \sin 2\theta = 0\). Hence proved that \(x^2 - 2x - \sin 2 \theta = y^2 + 2y - \sin 2 \theta = 0\)

Key Concepts

Differential Calculus

Differential calculus is a branch of mathematics that deals with the study of rates at which quantities change. It is primarily concerned with the derivative of a function, which represents an infinitesimal change in the function with respect to one of its variables. Understanding how to manipulate and use derivatives is crucial for solving complex mathematical problems.

In the context of the given exercise, while differential calculus isn't directly used, the ability to manipulate expressions and understand their rate of change in relation to trigonometric functions can offer deeper insight. When working with problems involving trigonometric identities, knowing how the derivatives of functions like \(\sin \theta\) and \(\cos \theta\) behave can assist in understanding how trigonometric expressions affect the overall solution.

For students preparing for exams, mastering differential calculus can improve problem-solving skills, enhance analytical thinking, and build a foundation for understanding more complex mathematical concepts.

IIT JEE Preparation

Preparing for the IIT JEE involves mastering a wide range of topics, including trigonometry and calculus. Understanding trigonometric identities and their applications is vital, as these topics often appear in various forms on the exam.

When solving problems like the given exercise, practicing the manipulation of trigonometric identities \(\sin^2 \theta + \cos^2 \theta = 1\) and understanding how to simplify complex expressions can considerably ease the solution process. This requires regular practice and a thorough understanding of fundamental concepts.

Here are some tips for mastering trigonometric equations in preparation for the IIT JEE:

• Regular practice with varying levels of difficulty in problems.
• A strong grasp of fundamental trigonometric identities.
• Application of differential calculus to understand function behavior.
• Breaking down complex problems into simpler parts for ease of solving.

By focusing on these methods, students can develop a robust mathematical foundation that will assist them in excelling in the IIT JEE.

Trigonometry Equations

Trigonometry equations involve a variety of identities and transformations that can simplify complex expressions. A standard identity used in trigonometry is \(\sin^2 \theta + \cos^2 \theta = 1\), which plays a significant role in simplifying expressions as shown in our exercise.

In the exercise, expressions for \(x\) and \(y\) were simplified using trigonometric identities. Mastering these identities involves understanding:

• How identities transform equations.
• The importance of simplifying expressions using known identities.
• Connecting different forms of trigonometric expressions to one another.

For instance, the expressions \(\sin \theta(1+\sin \theta) + \cos \theta(1+\cos \theta)\) and \(\sin \theta(1-\sin \theta) + \cos \theta(1-\cos \theta)\) were simplified using known trigonometric identities to solve the problem effectively. Practicing these transformations solidifies understanding and prepares students for tackling similar equations in exams.