Problem 103 - Mathematics for IIT JEE Main and Advanced Differential Calculus Algebra Trigonometry

Question

$$ \sin A \sin (A+2 B)-\sin B \sin (B+2 A)=\sin (A-B) \sin (A+B) . $$

Step-by-Step Solution

Answer

Using angle addition and sum-to-product formulas, we have verified the trigonometric identity: \[ \sin A \sin (A+2 B)-\sin B \sin (B+2 A)=\sin (A-B) \sin (A+B). \]

Step 1: Use the angle addition formula for sine on the left side

We will start by simplifying the left side of the equation. We'll use the angle addition formula for sine, which is given by: \[\sin(x+y) = \sin x \cos y + \cos x \sin y.\] Applying this formula, we have: \[\sin A \sin (A+2 B) - \sin B \sin (B+2 A) = \sin A(\sin A \cos 2B + \cos A \sin 2B) - \sin B (\sin B \cos 2A + \cos B \sin 2A).\]

Step 2: Expand and simplify the left side

Now, we will expand the left side and simplify the terms: \[\sin^2 A \cos 2B + \sin A \cos A \sin 2B - \sin^2 B \cos 2A - \sin B \cos B \sin 2A.\]

Step 3: Use the double-angle formulas

We'll now use the double-angle formulas for sine and cosine: \( \sin 2A = 2 \sin A \cos A .\) \( \sin 2B = 2 \sin B \cos B .\) We will substitute these in our equation: \[ \sin^2 A (1 - 2 \sin^2 B) + 2 \sin A \cos A (2 \sin B \cos B) - \sin^2 B (1 - 2 \sin^2 A) - 2 \sin B \cos B (2 \sin A \cos A). \]

Step 4: Simplify and factor the left side

Notice that the terms involving \(2 \sin A \cos A (2 \sin B \cos B)\) and \(2 \sin B \cos B (2 \sin A \cos A)\) cancel each other out. Simplifying the remaining terms gives: \[ \sin^2 A - 2 \sin^2 A \sin^2 B - \sin^2 B + 2 \sin^2 A \sin^2 B = \sin^2 A - \sin^2 B. \] We can factor the left side as a difference of squares: \[ (\sin A + \sin B)(\sin A - \sin B). \]

Step 5: Use the sum-to-product formulas

Now, we will use the sum-to-product formulas to transform the factored expression. The sum-to-product formula is given by: \[\sin x + \sin y = 2 \sin \frac{x+y}{2} \cos \frac{x-y}{2}.\] Applying this formula, we get: \[(\sin A + \sin B)(\sin A - \sin B) = 2 \sin \frac{A+B}{2} \cos \frac{A-B}{2} \cdot 2 \sin \frac{A-B}{2} \cos \frac{-(A-B)}{2}.\]

Step 6: Simplify and reach the desired result

Now, notice that \(\cos \frac{A-B}{2} = \cos \frac{-(A-B)}{2}\). So we can simplify the right side of the equation as: \[ 4 \sin \frac{A+B}{2} \cos \frac{A-B}{2} \sin \frac{A-B}{2} = \sin (A-B) \sin (A+B), \] which is the desired result. Thus, the original identity has been verified: \[ \sin A \sin (A+2 B)-\sin B \sin (B+2 A)=\sin (A-B) \sin (A+B). \]

Key Concepts

Angle Addition Formulas

Understanding trigonometric identities is essential for solving complex trigonometric equations. One of the fundamental formulas used in trigonometry is the angle addition formula. This formula helps in expressing the sine or cosine of a sum of angles, and it's given by:

• \( \sin(x+y) = \sin x \cos y + \cos x \sin y \)
  
• \( \cos(x+y) = \cos x \cos y - \sin x \sin y \)
  
  

In the original exercise, the angle addition formula for sine was applied to simplify the expression \( \sin A \sin (A+2B) - \sin B \sin (B+2A) \). By substituting into the equation using the angle addition formula, terms can be expanded and rearranged. This is the initial key step in transforming the left side of the given identity into a manageable form.
Understanding this formula makes it easier to break down and handle expressions involving trigonometric functions, especially when dealing with more complex or compounded angle relationships.

Double-angle Formulas

The double-angle formulas are a set of powerful tools used in trigonometry to deal with angles that are twice a given angle. These formulas are especially useful for simplifying expressions and solving equations. The commonly used double-angle formulas are:

• \( \sin 2A = 2 \sin A \cos A \)
  
• \( \cos 2A = \cos^2 A - \sin^2 A = 1 - 2 \sin^2 A = 2 \cos^2 A - 1 \)
  
  

In the provided solution, double-angle formulas were essential in further expanding and transforming the sine terms, specifically \( \sin 2A \) and \( \sin 2B \). By using these identities, we are able to replace the double angle terms with products of single angles, simplifying the expressions and identifying common terms that lead to simplification. Utilizing these formulas can make calculations more efficient and result in reduced complexity of mathematical problems involving trigonometry.

Sum-to-product Formulas

The sum-to-product formulas are another useful set of identities in trigonometry that transform the sum of trigonometric functions into products. This transformation can make certain types of equations easier to solve and manipulate. The key sum-to-product formulas include:

• \( \sin x + \sin y = 2 \sin \frac{x+y}{2} \cos \frac{x-y}{2} \)
  
• \( \sin x - \sin y = 2 \cos \frac{x+y}{2} \sin \frac{x-y}{2} \)
  
  

In the exercise solution, the sum-to-product formula was applied to the expression \( (\sin A + \sin B)(\sin A - \sin B) \). Using the sum-to-product identity, complex expressions are simplified by converting the sum of sines into products of sine and cosine. This allows for further manipulation, leading to the successful verification of the original identity. Mastering sum-to-product formulas equips you with a strategy to simplify and solve intricate trigonometric expressions efficiently.