Problem 18 - Mathematics for IIT JEE Main and Advanced Differential Calculus Algebra Trigonometry

Question

$$ \cot ^{4} A+\cot ^{2} A=\operatorname{cosec}^{4} A-\operatorname{cosec}^{2} A $$

Step-by-Step Solution

Answer

Rewriting the given identity in terms of sine and cosine, we get: \[\frac{\cos^{4} A}{\sin^{4} A}+\frac{\cos^{2} A}{\sin^2 A}= \frac{1}{\sin^{4} A} - \frac{1}{\sin^2 A}\] After manipulating the expression and using trigonometric identities, we simplify and rewrite the expression in terms of the original functions: \[\cot^2 A = \cosec^2 A - 1\] Hence, the trigonometric identity \(\cot ^{4} A+\cot ^{2} A=\operatorname{cosec}^{4} A-\operatorname{cosec}^{2} A\) is proven.

Step 1: Write the given identity in terms of sine and cosine

We are given the identity: \(\cot ^{4} A+\cot ^{2} A=\operatorname{cosec}^{4} A-\operatorname{cosec}^{2} A\) Recall that, \(\cot A = \frac{\cos A}{\sin A}\) and \(\cosec A = \frac{1}{\sin A}\) So, we can rewrite the given identity as: \[\frac{\cos^{4} A}{\sin^{4} A}+\frac{\cos^{2} A}{\sin^2 A}= \frac{1}{\sin^{4} A} - \frac{1}{\sin^2 A}\]

Step 2: Manipulate the expression

Now let's work with the expression: \[\frac{\cos^{4} A}{\sin^{4} A}+\frac{\cos^{2} A}{\sin^2 A}= \frac{1}{\sin^{4} A} - \frac{1}{\sin^2 A}\] To add and subtract these fractions, we need a common denominator. The common denominator is: \(\sin^{4} A\) We express each term with this common denominator and simplify: \[\frac{\cos^{4} A+\cos^{2} A\sin^{2} A}{\sin^{4} A} = \frac{\sin^{4} A - \sin^{2} A}{\sin^{4} A}\]

Step 3: Factor and simplify both sides

Now, let's factor both sides of the expression: \[\frac{\cos^{2} A(\cos^{2} A + \sin^{2} A)}{\sin^{4} A} = \frac{\sin^{2} A(\sin^{2} A - 1)}{\sin^{4} A}\] Using the identity: \(\cos^{2} A + \sin^{2} A = 1\), we can simplify the left-hand side: \[\frac{\cos^{2} A(1)}{\sin^{4} A} = \frac{\sin^{2} A(\sin^{2} A - 1)}{\sin^{4} A}\] Now, we can cancel \(\sin^{2} A\) from both sides: \[\frac{\cos^{2} A}{\sin^{2} A} = \frac{\sin^{2} A - 1}{\sin^{2} A}\]

Step 4: Rewrite the expression in terms of original functions

Now that we have a simplified expression, let's rewrite it in terms of the original functions. \[\cot^2 A = \cosec^2 A - 1\] Thus, we've proven the trigonometric identity: \[\cot ^{4} A+\cot ^{2} A=\operatorname{cosec}^{4} A-\operatorname{cosec}^{2} A\]

Key Concepts

cotangent

The cotangent function is an important trigonometric function often represented as \( \cot A \). It's defined as the reciprocal of the tangent function. In terms of sine and cosine, cotangent is expressed as:

• \( \cot A = \frac{\cos A}{\sin A} \)

This identity shows the relationship between the cotangent, cosine, and sine functions. It's particularly useful when trying to simplify expressions that involve angles in a right triangle or the unit circle.
Understanding the cotangent function aids in solving various trigonometric equations and simplifying expressions involving other trigonometric identities.
By rewriting the cotangent in terms of sine and cosine, it becomes easier to manipulate and find common ground between different trigonometric expressions.

cosecant

In trigonometry, the cosecant function, denoted as \( \cosec A \), is a less common but equally crucial function. It's essentially the reciprocal of the sine function:

• \( \cosec A = \frac{1}{\sin A} \)

This definition highlights its role in relating to the sine function. Though not as commonly used as sine or cosine, the cosecant function often appears when simplifying complex expressions and equations.
In the context of the identity \( \cot^4 A+\cot^2 A=\operatorname{cosec}^4 A-\operatorname{cosec}^2 A \), rewriting in terms of sine reveals new ways to reduce the complexity of the expression, making problems easier to solve.

simplifying expressions

Simplifying trigonometric expressions is about breaking down complex equations into simpler, more manageable parts. This process involves using fundamental trigonometric identities and algebraic manipulation.
To simplify expressions like \( \frac{\cos^{4} A}{\sin^{4} A} + \frac{\cos^{2} A}{\sin^2 A} \), one can combine fractions with a common denominator, factor expressions, and use identities. For this equation, the identity \( \cos^{2} A + \sin^{2} A = 1 \) plays a significant role.
By understanding these techniques, you can approach simplifying expressions with confidence, converting them into forms that are easier to analyze or equate with other expressions.

trigonometric equations

Solving trigonometric equations often requires converting complex expressions into simpler forms using trigonometric identities. The goal is to find values that satisfy equations like \( \cot ^{4} A+\cot ^{2} A=\operatorname{cosec}^{4} A-\operatorname{cosec}^{2} A \).
To approach such problems, rewriting trigonometric functions in terms of sine and cosine can help. From there, simplifying using identities and algebraic steps, like factoring or finding common denominators, becomes manageable.
Successfully tackling trigonometric equations involves understanding the behaviors and properties of trigonometric functions, allowing precise manipulation towards a solution.