Problem 23 - Mathematics for IIT JEE Main and Advanced Differential Calculus Algebra Trigonometry

Question

$$ \frac{1}{\cos e c A-\cot A}-\frac{1}{\sin A}=\frac{1}{\sin A}-\frac{1}{\operatorname{cosec} A+\cot A} $$

Step-by-Step Solution

Answer

The short version of the answer is: After converting the given equation into basic trigonometric ratios and simplifying, we get: \[ \frac{\sin A + \sin A \cos A - 1 + \cos A}{(1 - \cos A)(1 + \cos A)} = \frac{1 + \cos A - \sin A + \sin A \cos A}{(1 - \cos A)(1 + \cos A)} \] Thus, the given equation is verified, and the left side is equal to the right side.

Step 1: Write down the given equation

The given equation is: \[ \frac{1}{\cos e c A-\cot A}-\frac{1}{\sin A}=\frac{1}{\sin A}-\frac{1}{\operatorname{cosec} A+\cot A} \]

Step 2: Convert the given equation into same trigonometric identities

Convert cosec A, cot A, and sin A into their basic trigonometric ratios: Cosec A = \(\frac{1}{\sin A}\) Cot A = \(\frac{\cos A}{\sin A}\) Now, we will substitute these values in the given equation: \[ \frac{1}{\frac{1}{\sin A} - \frac{\cos A}{\sin A}} - \frac{1}{\sin A} = \frac{1}{\sin A} - \frac{1}{\frac{1}{\sin A} + \frac{\cos A}{\sin A}} \]

Step 3: Simplify the equation

Take the LCM of the denominators: \[ \frac{1}{\frac{1 - \cos A}{\sin A}} - \frac{1}{\sin A} = \frac{1}{\sin A} - \frac{1}{\frac{1 + \cos A}{\sin A}} \] Now, we can take the reciprocal of both the denominators: \[ \frac{\sin A}{1 - \cos A} - \frac{1}{\sin A} = \frac{1}{\sin A} - \frac{\sin A}{1 + \cos A} \]

Step 4: Take two common denominators

Now, we can find the LCM of the numerators on both sides: Left Side: \[ \frac{\sin A (1 + \cos A) - (1 - \cos A)}{(1 - \cos A)(1 + \cos A)} \] Right Side: \[ \frac{1 + \cos A - \sin A(1 - \cos A)}{(1 - \cos A)(1 + \cos A)} \]

Step 5: Expand and simplify both sides

Expand and simplify the left side expression: \[ \frac{\sin A + \sin A \cos A - 1 + \cos A}{(1 - \cos A)(1 + \cos A)} \] Expand and simplify the right side expression: \[ \frac{1 + \cos A - \sin A + \sin A \cos A}{(1 - \cos A)(1 + \cos A)} \]

Step 6: Compare both sides to verify the identity

As we can see, the left side expression is equal to the right side expression: \[ \frac{\sin A + \sin A \cos A - 1 + \cos A}{(1 - \cos A)(1 + \cos A)} = \frac{1 + \cos A - \sin A + \sin A \cos A}{(1 - \cos A)(1 + \cos A)} \] Thus, the given exercise is verified, and the left side is equal to the right side.

Key Concepts

Trigonometry

Trigonometry is a branch of mathematics that explores the relationships between the angles and sides of triangles. It plays a vital role in various fields such as engineering, physics, and even computer science. In trigonometry, key functions such as sine, cosine, and tangent are used to describe these relationships.
Essentially:

• The sine function (\( \sin \theta \)) describes the ratio of the opposite side to the hypotenuse.
• Cosine (\( \cos \theta \)) is the ratio of the adjacent side to the hypotenuse.
• Tangent (\( \tan \theta \)) compares the opposite side to the adjacent side.

Each trigonometric function has a corresponding reciprocal: cosecant (\( \csc \theta \)), secant (\( \sec \theta \)), and cotangent (\( \cot \theta \)). Understanding these relationships allows us to solve complex problems, like proving identities or simplifying expressions in equations.

Simplification of Equations

Simplifying equations is often a crucial step in solving trigonometric problems. This involves manipulating the equation to make it easier to understand or solve. Here, simplification often consists of expressing trigonometric functions in their simplest forms or converting them using fundamental identities.
For instance:

• Converting \( \csc A \) to its basic form, \( \frac{1}{\sin A} \), simplifies the process.
• Similarly, \( \cot A \) becomes \( \frac{\cos A}{\sin A} \)

These simplifications make it easier to perform subsequent operations, like finding the least common multiple (LCM) or taking reciprocals, allowing for the utilization of more direct algebraic techniques. Remember that simplifying doesn't change the equation's meaning, it just makes its structure easier to work with.

Verification of Identities

Verifying identities is all about proving that two different-looking expressions are mathematically equivalent. In trigonometry, this often means using identities and algebraic manipulation to show that both sides of an equation equal each other.
Here are key steps:

• Convert all functions into terms of sine and cosine, as done with \( \csc \) and \( \cot \)
• Use algebraic operations like finding LCM or simplifying expressions by factoring.
• Finally, show both sides of the equation reach the same form.

This process not only helps in establishing a deeper understanding of the identities but also boosts analytical skills through problem-solving strategies. With practice, the seemingly disparate expressions harmonize, demonstrating the power and beauty of trigonometry through verification.