Problem 115 - Mathematics for IIT JEE Main and Advanced Differential Calculus Algebra Trigonometry

Question

$$ y=\frac{\sin ^{2} x}{1+\cot x}+\frac{\cos ^{2} x}{1+\tan x} $$

Step-by-Step Solution

Answer

The simplified answer to the given trigonometric expression is \(y = (\sin(x) + \cos(x))(1 - \sin(x)\cos(x))\).

Step 1: Recognize the identities

First, let's recognize important trigonometric identities. Remember \(\sin^2(x) + \cos^2(x) = 1\) , \(\cot(x) = \frac{1}{\tan(x)}\) and \(\tan(x) = \frac{1}{\cot(x)}\)

Step 2: Replace cot(x) and tan(x) in denominators

Now, replace \(\cot(x)\) with \(\frac{1}{\tan(x)}\) in the denominator of the first fraction, and replace \(\tan(x)\) with \(\frac{1}{\cot(x)}\) in the second fraction. The equation thus becomes: \[y = \frac{\sin^2(x)}{1+\frac{1}{\tan(x)}} + \frac{\cos^2(x)}{1+\frac{1}{\cot(x)}}\]

Step 3: Simplify the fractions

Next, let's make the denominator a single fraction for both terms: \[y = \frac{\sin^2(x)}{\frac{\tan(x)+1}{\tan(x)}}+\frac{\cos^2(x)}{\frac{\cot(x)+1}{\cot(x)}}\] or \[y = \sin^2(x) \cdot \tan(x) + \cos^2(x) \cdot \cot(x)\]

Step 4: Substitute tan and cot with sin and cos

Now replace \(\tan(x)\) with \(\frac{\sin(x)}{\cos(x)}\) and \(\cot(x)\) with \(\frac{\cos(x)}{\sin(x)}\). It gives: \[y = \sin^3(x) + \cos^3(x)\]

Step 5: Recognize the pattern

Now let's observe that \(\sin^3(x) + \cos^3(x)\) can be factored into \((\sin(x) + \cos(x))(\sin^2(x) - \sin(x)\cos(x) + \cos^2(x))\)

Step 6: Substitute sin^2(x) + cos^2(x)

Remembering that \(\sin^2(x) + \cos^2(x) = 1\), the equation can be simplified to: \[y = (\sin(x) + \cos(x))(1 - \sin(x)\cos(x))\]

Key Concepts

Trigonometric Functions

Trigonometric functions form the foundation of trigonometry, and they are widely used in various mathematical equations. In this exercise, we primarily deal with two essential functions - sine (\(\sin(x)\)) and cosine (\(\cos(x)\)). These functions relate the angles of a right triangle to its sides. They are crucial for identities like \(\sin^2(x) + \cos^2(x) = 1\).
This identity is fundamental because it allows simplification in complex expressions, transforming equations into more manageable forms. Additionally, the tangent (\(\tan(x)\)) and cotangent (\(\cot(x)\)) functions offer other means to express these relationships. Deriving from sine and cosine, \(\tan(x)\) is expressed as \(\frac{\sin(x)}{\cos(x)}\) and similarly, \(\cot(x)\) is \(\frac{\cos(x)}{\sin(x)}\). Knowing these functions helps you manipulate and rearrange trigonometric expressions effectively.

Algebraic Manipulation

Algebraic manipulation involves rewriting expressions in different forms to simplify or solve equations. In this problem, we use algebraic manipulation to replace complex terms with simpler expressions involving trigonometric identities and functions. For instance, \(\cot(x)\) is substituted with \(\frac{1}{\tan(x)}\) and vice versa, allowing us to consolidate the denominators into single fractions.
Through algebraic manipulation, fractions like \(1 + \frac{1}{\tan(x)}\) can be rewritten, ultimately transforming complex fractions into multiplication forms, such as \(\sin^2(x) \cdot \tan(x)\). This transformation reduces the complexity of equations and helps reveal essential patterns or identities hidden in a problem. By mastering algebraic techniques, you can effectively simplify and solve subtle trigonometric challenges.

Simplification Techniques

Simplification techniques are crucial for distilling complex mathematical expressions into simpler, more manageable forms. In solving this exercise, recognizing and utilizing trigonometric identities like \(\sin^2(x) + \cos^2(x) = 1\) is key. This specific identity allows the expression to be broken down into its components, facilitating a step-by-step simplification.
Combining like terms and factoring expressions are typical techniques used. For instance, \(\sin^3(x) + \cos^3(x)\) can be factored into \((\sin(x) + \cos(x))(1 - \sin(x)\cos(x))\). Such methods help both in recognizing simpler expressions within a larger equation and in ensuring each part of the equation is accounted for when reducing it to its most straightforward form.

Function Transformation

Function transformation involves altering a function's input or output to simplify it or fit a specific format. During the exercise, you transform the functions by changing the terms, making them easier to work with or visualize. Specifically, moving from expressions involving tangent and cotangent to those involving sine and cosine plays a vital role.
This transformation is crucial for simplifying the equation, as it allows the problem to be tackled using fundamental trigonometric identities, minimizing complexity. Understanding how to interchange between \(\tan(x)\) and \(\frac{\sin(x)}{\cos(x)}\) (or between \(\cot(x)\) and \(\frac{\cos(x)}{\sin(x)}\)) reduces complicated functions into their basic trigonometric components.