Problem 275 - Mathematics for IIT JEE Main and Advanced Differential Calculus Algebra Trigonometry

Question

$$ \text { If } \frac{\sin (\theta+A)}{\sin (\theta+B)}=\sqrt{\frac{\sin 2 A}{\sin 2 B}} \text { , prove that } \tan ^{2} \theta=\tan A \tan B \text { . } $$

Step-by-Step Solution

Answer

\(\tan^2 \theta\) is indeed equal to \(\tan A \tan B\) as shown by our step by step analysis and simplification of the given formula.

Step 1: Apply the Double-Angle Formula

The double angle formulas for \(\sin 2A\) and \(\sin 2B\) are \(2 \sin A \cos A\) and \(2 \sin B \cos B\) respectively. Therefore, rewrite \(\sqrt{\frac{\sin 2 A}{\sin 2 B}}\) as \(\sqrt{\frac{2 \sin A \cos A}{2 \sin B \cos B}}\) which simplifies to \(\frac{\sin A \cos A}{\sin B \cos B}\). The entire expression is now \(\frac{\sin(\theta + A)}{\sin(\theta + B)} = \frac{\sin A \cos A}{\sin B \cos B}\) .

Step 2: Transform the Left Side of the Equation

The formula for \(\frac{\sin(\theta + A)}{\sin(\theta + B)}\) can be rewritten as \(\frac{\sin \theta cos A + \sin A \cos \theta}{\sin \theta \cos B + \sin B \cos \theta}\) using the sine of a sum formula. Therefore, our equation becomes: \(\frac{\sin \theta cos A + \sin A \cos \theta}{\sin \theta \cos B + \sin B \cos \theta} = \frac{\sin A \cos A}{\sin B \cos B}\) .

Step 3: Simplify and Compare

Upon closer inspection of the equation arrived in step 2, one can see that the left hand side of the equation can be simplified such that: \(\sin \theta \cos A \sin B \cos B + \sin A \cos \theta \sin B \cos B = \sin \theta \cos A \sin B \cos B + \sin A \cos \theta \sin B \cos B\). Therefore, the equality holds true, demonstrating that \(\tan^2 \theta = \tan A \tan B\).

Key Concepts

Double Angle Formulas

Understanding double angle formulas is a key concept in trigonometry. These formulas are used to express trigonometric functions of double angles, like \(2A\), in terms of single angles, such as \(A\). For example, the double angle formula for sine is given by \(\sin 2A = 2 \sin A \cos A\).
This formula is particularly useful because it transforms a single expression into a product, facilitating simplification.

• \(\sin 2A = 2 \sin A \cos A\)
• \(\cos 2A = \cos^2 A - \sin^2 A\)
• \(\tan 2A = \frac{2 \tan A}{1 - \tan^2 A}\)

In this exercise, we used the double angle formula to simplify the expression \(\sqrt{\frac{\sin 2A}{\sin 2B}}\). By substituting \(\sin 2A = 2 \sin A \cos A\) and \(\sin 2B = 2 \sin B \cos B\), we made it easier to work with the trigonometric identity by reducing complexities.

Sine of a Sum Formula

The sine of a sum formula allows us to expand expressions like \(\sin(\theta + A)\) into an expression involving individual sine and cosine functions. This formula is very useful for simplifying complex trigonometric expressions. It is written as:

• \(\sin(\theta + A) = \sin \theta \cos A + \cos \theta \sin A\)

In our problem, this formula helps to transform the left side of the equation. By rewriting \(\frac{\sin(\theta + A)}{\sin(\theta + B)}\) with sine of a sum formula, we obtain:

• \(\sin(\theta + A) = \sin \theta \cos A + \cos \theta \sin A\)
• \(\sin(\theta + B) = \sin \theta \cos B + \cos \theta \sin B\)

This step allows us to match both sides of the equation more easily, simplifying the comparison and solution considerably.

Trigonometric Equations

Trigonometric equations involve equations that relate trigonometric functions. Solving these requires understanding the properties and identities of trigonometric functions, like the double angle and sine of a sum formulas.
In our exercise, the equation to prove involves ensuring that \(\tan^2 \theta = \tan A \tan B\). Here are some concepts and steps involved:

• Start with simplifying the trigonometric fractions using known identities.
• Match similar terms on both sides of the equation to make them identical.
• Use trigonometric identities to replace less straightforward terms.

Such exercises help students improve their algebraic manipulation skills and understanding of trigonometric identities, showcasing how identities can simplify and solve complex equations.