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

Question

$$ \sin A \sin B \sin (A-B)+\sin B \sin C \sin (B-C)+\sin C \sin A \sin (C-A)+\sin (A-B) \sin (B-C) \sin (C-A)=0 $$

Step-by-Step Solution

Answer

After recognizing the pattern in the terms we apply the sine subtraction formula to the arguments of the sine function in each term. Doing so, we get a sum of terms where each of them cancels each other, thus validating the expression to be equal to zero.

Step 1: Recognize the pattern

Notice the symmetry between the three terms. In each term, the first two arguments of the sine functions add up to the third one's. This is a hint that could lead to a simplification when using the sine subtraction formula \(\sin(A-B) = \sin A \cos B - \cos A \sin B\).

Step 2: Apply the sine subtraction formula

Firstly apply the sine subtraction formula to the arguments in each of the terms. We get \(\sin A \sin B (\sin A \cos B - \cos A \sin B) + \sin B \sin C (\sin B \cos C - \cos B \sin C) + \sin C \sin A (\sin C \cos A - \cos C \sin A)\). Now if we expand each term, what we notice is that there will be cancellation of terms.

Step 3: Simplify

After expanding, we get \( \sin^2 A \sin B \cos B - \sin A \cos A \sin^2 B + \sin^2 B \sin C \cos C - \sin B \cos B \sin^2 C + \sin^2 C \sin A \cos A - \sin C \cos C \sin^2 A\). Now, we see that the terms cancel each other, reducing the expression to zero.

Key Concepts

Symmetric Functions

Symmetric functions are those that remain unchanged under certain transformations like swapping variables. In the original exercise, notice how the expression involves cyclical terms.

• Each term includes different combinations of \( \sin A, \sin B, \) and \( \sin C \).
• These combinations are rotated positions in similar structural forms.
• This repetitive pattern hints at potential simplifications.

Recognizing symmetry helps predict how terms might interact or cancel.
When studying trigonometric identities, always look for symmetry to simplify complex expressions.
This will often lead to easier manipulation of terms and help streamline problem-solving.

Sine Subtraction Formula

The Sine Subtraction Formula allows you to break down expressions like \( \sin(A-B) \), making it simpler to handle. The formula is given by:

• \( \sin(A-B) = \sin A \cos B - \cos A \sin B \)
• It transforms a subtraction within the sine function into a manageable form.

In our exercise, applying this formula helped express terms with separate sine and cosine parts.
This transformation makes it easier to spot cancellations after multiplying out the expressions.
Always remember this formula when dealing with trigonometric expressions involving subtraction or addition.
It serves as a powerful tool in reducing complex trigonometric equations.

Simplification Techniques

Simplification in mathematics involves reducing complex expressions into simpler forms. Here, we focused on using identities and algebraic manipulations.

• First, apply known identities like the Sine Subtraction Formula.
• Next, expand the expressions to reveal terms that cancel out.

As seen in step 3 of the original solution, many terms canceled out naturally when expanded.
This left us with zero, proving that the initial complex equation was indeed symmetrical and balanced.
Mastering simplification techniques can greatly ease solving any mathematical problem by transforming them into manageable pieces.
These techniques are crucial not just in trigonometry but across all areas of mathematics.