Problem 285 - Mathematics for IIT JEE Main and Advanced Differential Calculus Algebra Trigonometry

Question

$$ \text { Prove that } \sin ^{2}(\theta+\alpha)+\sin ^{2}(\theta+\beta)-2 \cos (\alpha-\beta) \sin (\theta+\alpha) \sin (\theta+\beta) \text { is independent of } \theta \text { . } $$

Step-by-Step Solution

Answer

The given expression is indeed independent of \(\theta\). After applying trigonometric identities and simplifying the resulting expression, all instances of \(\theta\) cancel out.

Step 1: Apply Trigonometric Identities

To start, apply the identities for sine and cosine of two angles, expressed as \(\sin(x+y)=\sin x \cos y + \cos x \sin y\) and \(\cos(x-y)= \cos x \cos y + \sin x \sin y \) to the terms in the expression. So the expression \(\sin ^{2}(\theta+\alpha)+\sin ^{2}(\theta+\beta)-2 \cos(\alpha-\beta) \sin (\theta+\alpha) \sin (\theta+\beta)\) will become \([(\sin\theta\cos\alpha + \cos\theta\sin\alpha)^2] + [(\sin\theta\cos\beta + \cos\theta\sin\beta)^2] - 2[(\cos\alpha\cos\beta+\sin\alpha\sin\beta)][(\sin\theta\cos\alpha + \cos\theta\sin\alpha)][(\sin\theta\cos\beta + \cos\theta\sin\beta)]\).

Step 2: Expand and Simplify

Next, expand the squares and other multiplications such that every term is a product of sine and/or cosine functions. Also, make sure to distribute the negative sign that precedes the third term. The expression now becomes a long polynomial, but it can be simplified by combining like terms: terms that have the same functions of the same angles.

Step 3: Cancel Out Theta

Combine terms which have similar components like term having \(\sin\theta \cos\theta\) and look for opportunities to cancel out \(\sin\theta\) or \(\cos\theta\). Keep simplifying and combining until all instances of \(\sin\theta\) and \(\cos\theta\) have been cancelled.

Step 4: Conclusion

If all instances of \(\sin\theta\) and \(\cos\theta\) have been cancelled, it means that the original expression is independent of \(\theta\).

Key Concepts

Sine and Cosine Functions

Sine and cosine are fundamental trigonometric functions that relate the angles and sides of right triangles. They also describe oscillations, like waves, and have values that range between -1 and 1.

- The sine function, denoted as \( \sin \theta \), corresponds to the y-coordinate or the length of the side opposite the angle in a unit circle.- The cosine function, denoted as \( \cos \theta \), relates to the x-coordinate or the length of the adjacent side in a unit circle.

These functions are periodic, meaning they repeat their values in cycles. Key properties include:

• Sine and cosine have a period of \( 2\pi \). This means their graphs repeat every \( 2\pi \) radians.
• The critical values of sine and cosine are found at specific angles: 0, \( \pi/2 \), \( \pi \), \( 3\pi/2 \), and \( 2\pi \). These correspond to the maximum, minimum, and zero points on their graphs.

Understanding sine and cosine functions is crucial for solving more complex trigonometric problems, including those involving angle addition.

Angle Addition Formulas

The angle addition formulas are essential tools in trigonometry, allowing us to find the sine or cosine of the sum or difference of two angles. They're crucial for transforming expressions into more manageable forms.

Here are the key angle addition formulas:

• The sine addition formula: \( \sin(x+y) = \sin x \cos y + \cos x \sin y \)
• The cosine addition formula: \( \cos(x-y) = \cos x \cos y + \sin x \sin y \)

These formulas reflect how trigonometric functions behave with angle adjustments, giving us a way to decompose complex expressions into simpler parts.

By using the angle addition formulas, you can break down expressions such as \( \sin^2(\theta+\alpha) \). This makes it easier to manage and ultimately simplifies to check for independence from specific variables, like \( \theta \) in the given problem.

Simplification of Trigonometric Expressions

Simplifying trigonometric expressions is a critical skill, allowing us to reduce expressions to a more workable form. This can involve cancellations, factorizations, and recognizing common patterns or identities.

To simplify expressions like \( \sin^2(\theta+\alpha) + \sin^2(\theta+\beta) - 2\cos(\alpha-\beta)\sin(\theta+\alpha)\sin(\theta+\beta) \), follow these steps:

• Apply known identities: Use angle addition identities to express complex angles using basic sine and cosine functions.
• Expand and distribute: Carefully expand squares and distribute terms. Pay attention to signs.
• Combine and cancel: Identify terms with the same trigonometric functions and combine them. Look for cancellations to reduce complexity, such as those involving \( \sin \theta \) and \( \cos \theta \).

The goal in simplification is to arrive at an expression that shows the desired property, like independence of a variable. Each step builds upon the previous, and their proper execution ensures correctness. In this problem, simplifying demonstrates that the expression indeed does not depend on \( \theta \), achieving the proof required.