Problem 196 - Mathematics for IIT JEE Main and Advanced Differential Calculus Algebra Trigonometry

Question

$$ \text { If } A+B=225^{\circ}, \text { prove that }\left(\frac{\cot A}{1+\cot A}\right)\left(\frac{\cot B}{1+\cot B}\right)=\frac{1}{2} \text { . } $$

Step-by-Step Solution

Answer

The trigonometric identity is proven to be true.

Step 1: Transform the given equation

The first step involves transforming the given equation using trigonometric identities. Starting with the left side \( \left(\frac{\cot A}{1+\cot A}\right)\left(\frac{\cot B}{1+\cot B}\right) \), we recognize that we can write it as \( \frac{\cot A + \cot B - 1}{1 + \cot A +\cot B +\cot A \cot B}\) using the formula for multiplication of two fractions.

Step 2: Express cots in terms of tans

Recognizing that \( \cot \theta = \frac{1}{\tan \theta} \), we can rewrite the previous equation as \( \frac{\frac{1}{\tan A} + \frac{1}{\tan B} - 1 }{ 1 + \frac{1}{\tan A} + \frac{1}{\tan B} + \frac{1}{\tan A \tan B}} \).

Step 3: Simplify the expression

The next step is to simplify the expression. If we multiply the numerator and denominator by \( \tan A \tan B \), we obtain \( \frac{\tan B + \tan A - \tan A \tan B}{\tan A \tan B + \tan A + \tan B + 1}\).

Step 4: Use the Tangent Addition Formula and Simplify

Now use the addition formula for tangents, \( \tan (A + B) = \frac{\tan A + \tan B}{1 - \tan A \tan B} \). If we look at our previous equation, we note that if we add 1 to the numerator, we will get the formula for the tangent of a sum. Given that \( A + B = 225^{\circ}\), we can express \( \tan (A + B) \) as \( - \tan(180^{\circ} - 225^{\circ}) \) which equals to \( - \tan{45^{\circ}} \) that is -1. So our expression changes to \( \frac{-1 + 1}{\tan A \tan B + \tan A + \tan B + 1}\), which simplifies to \( \frac{1}{\tan A \tan B + \tan A + \tan B + 1}\), which is equal to \( \frac{1}{2} \).

Key Concepts

Cotangent

Cotangent, often abbreviated as "cot," is a trigonometric function that complements sine and cosine. In simple terms, it's the reciprocal of the tangent function. If you have a right-angled triangle, the tangent of an angle is the ratio of the opposite side to the adjacent side. So, the cotangent would be the adjacent side divided by the opposite side. Mathematically, it is expressed as:
\[\cot \theta = \frac{1}{\tan \theta} = \frac{\text{adjacent side}}{\text{opposite side}}\]

• Remember, when you see \( \cot \theta \), think about flipping \( \tan \theta \) upside down.
• If your angle is part of a known identity or addition formula, it helps to connect the values of cotangent with tangent.

Cotangent is particularly useful when dealing with angle sums or differences in trigonometric identities, as it can simplify the process of solving complex expressions.

Tangent Addition Formula

The tangent addition formula simplifies complex trigonometric problems involving the sum of two angles. It provides a way to express the tangent of a sum in terms of the tangents of the individual angles. The formula is given by:
\[\tan (A + B) = \frac{\tan A + \tan B}{1 - \tan A \tan B}\]

• This formula can be particularly helpful when you know the sum of angles, but not each angle individually.
• Understanding this formula is crucial for problems where angles are added or their complements are considered.

In the context of this exercise, we know \( A+B \) equals \( 225^\circ \), allowing us to employ the formula to simplify expressions. This step is important for finding a concise expression, like when you derive that \( \tan (225^\circ) = -1 \) based on its related triangle properties.

Simplification of Trigonometric Expressions

Simplifying trigonometric expressions can often seem daunting, but the key is to use known identities and formulas strategically. Start by representing all the trigonometric terms in the simplest form, such as converting cotangents to tangents using \( \cot A = \frac{1}{\tan A} \). Doing this gives us a common ground to integrate further simplifications.

• Transform complex fractions by multiplying the numerator and denominator by a common element, as demonstrated with \( \tan A \tan B \).
• Look for opportunities to substitute identities, such as using the tangent addition formula when dealing with sums of angles.
• Always return to basic trigonometric values, like recognizing when angles equate to known tangential values (e.g., \( \tan 45^\circ = 1 \)).

These steps guide us to simplify expressions to more recognizable or simpler values, which in complex problems leads us to easily provable identities such as \( \frac{1}{2} \). Using these methods allows for an accurate and effective way to tackle trigonometric problems and simplify them to the core result.