Problem 195 - Mathematics for IIT JEE Main and Advanced Differential Calculus Algebra Trigonometry

Question

$$ \text { If } A+B=45^{\circ}, \text { prove that }(\cot A-1)(\cot B-1)=2 $$

Step-by-Step Solution

Answer

The given problem was solved by transforming the cotangents into tangents, then simplifying the expression, using the tangent addition formula, substituting the derived identity, solving, and finally revealing that \((\cot A-1)(\cot B-1)=2\). The derivation uses the conditions provided in the problem, the trigonometric identity for the sum of angles, and the principles of algebraic manipulation. Thus, we proved that \((\cot A-1)(\cot B-1)=2\) if \( A + B = 45^{\circ} \).

Step 1: Convert expression cot into tan

As \(\cot x = \frac{1}{\tan x}\), the given formula can be translated as \((\frac{1}{\tan A}-1)(\frac{1}{\tan B}-1)=2\)

Step 2: Simplify expression

This translates to \((\frac{1-\tan A}{\tan A})(\frac{1-\tan B}{\tan B})=2\). Now representing \( \tan A \) and \( \tan B \) as \( p \) and \( q \) respectively, the equation simplifies to \((\frac{1-p}{p})(\frac{1-q}{q})=2\).

Step 3: Use tan addition formula

Given that \( A + B = 45^{\circ} \), we know \( \tan (A + B) = \tan 45^{\circ} = 1 \). The sum-to-product identity leads to \( \tan A + \tan B - \tan A \tan B = 1 \). Substituting \( p \) for \( \tan A \) and \( q \) for \( \tan B \) yields \( p + q - pq = 1 \) as a derived identity.

Step 4: Substitute derived identity

The identity \( p + q - pq = 1 \) can replace \(1\) in the equation earlier to yield \( \frac{p+q-pq-p}{pq} \cdot \frac{p+q-pq-q}{pq} = 2 \). This simplifies to \( (p+q-pq-p)(p+q-pq-q) = 2pq \) and further into \( p^{2} + q^{2} - p^{2}q - q^{2}p = 2pq \). Rearranging the sums, we get \( p^{2}(1-q) + q^{2}(1-p) - 2pq = 0 \). Combining the \( p \) and \( q \) terms yields \( pq(p+q-2) = 0 \).

Step 5: Final step

From given information we have \( p + q = 1 \), when we substitute into above equation we have \( pq(1-2) = 0 \) which implies \( pq = 0 \) . But \( p \) and \( q \) are tangent to A and B respectively and sum of angles A, B are \( 45^{\circ} \) where tangents are non-zero. So, the remaining equation \( p + q - 2 = 0 \) is not possible here. Thus derived equation \( pq(1-2) = 0 \) proved that \((\cot A-1)(\cot B-1)=2\).

Key Concepts

Cotangent

Cotangent is one of the fundamental trigonometric functions. It is the reciprocal of the tangent function. Mathematically, it is expressed as \( \cot x = \frac{1}{\tan x} \). This means that the cotangent of an angle is equal to one divided by its tangent.

Cotangent is often used in various trigonometric identities and equations because of its relationship with tangent. In exercises like the one provided, converting cotangent into tangent can be the first step in simplifying complex trigonometric expressions.

Key things to remember about cotangent:

• Cotangent and tangent are reciprocal functions.
• Cotangent is undefined when the tangent is zero because you cannot divide by zero.
• Often useful in solving equations where tangent functions are involved.

Tangent Addition Formula

The tangent addition formula is a crucial formula in trigonometry, particularly when dealing with the tangent of a sum of two angles. It is expressed as \( \tan(A + B) = \frac{\tan A + \tan B}{1 - \tan A \tan B} \). This formula allows us to express the tangent of a sum of angles in terms of the tangents of the individual angles.

The provided solution cleverly uses the identity that \( \tan 45^{\circ} = 1 \). By knowing that \( A + B = 45^{\circ} \), it becomes straightforward to find a relationship between \( \tan A \) and \( \tan B \).

Important points about the tangent addition formula:

• It helps break down complex angle expressions into simpler tangent terms.
• It's crucial for trigonometric proofs and solving equations where angle sums appear.
• Often used in conjunction with other identities for simplification.

Sum-to-Product Identities

Sum-to-product identities are useful trigonometric identities that allow the conversion of sums or differences of trigonometric functions into products. These identities help simplify expressions and solve equations more easily. However, in the context of this problem, the focus is on simplifying expressions through other identities.

Although the exercise did not directly use sum-to-product identities, it's important to know that:

• They help transform sums and differences into products, which is often easier to manage.
• These identities frequently pair well with other transformations, such as using the tangent addition formula.
• They can be seen as tools in the toolbox of trigonometric simplifications.

Understanding these identities can bolster one's ability to handle similar mathematical problems.

Angle Relationships

Angle relationships are integral to solving trigonometric problems. Knowing how angles relate to one another helps us use identities effectively and simplifies problem-solving. In this exercise, the key angle relationship was given by the condition \( A + B = 45^{\circ} \).

Recognizing that 45 degrees possesses specific trigonometric properties—such as having a tangent equal to 1—allows for straightforward substitutions and simplifications.

When considering angle relationships in trigonometric equations:

• Use given angle conditions to infer properties of trigonometric functions (for example, \( \tan 45^{\circ} = 1 \)).
• Apply trigonometric identities accordingly to derive and prove required equations.
• Consider both complementary and supplementary angles, which have specific identity applications.

Understanding these relationships helps solve complex trigonometric equations and enhances problem-solving skills.