Let α, β be such that π

Q: Why do we square the equations? Isn't that just making it more complicated?

It feels like we are making the numbers larger, but we are actually performing a strategic 'elimination' maneuver. By squaring, we create the form $\sin^2 \theta + \cos^2 \theta$, which is the most powerful tool in trigonometry because it collapses two variables into the constant 1. It is not about the size of the numbers; it is about the structural simplification.

Q: Why did we reject the positive root?

This is the classic JEE trap! We are given the constraint $\pi < \alpha - \beta < 3\pi$. When we divide this by 2 to find the range of our target angle, we get $\frac{\pi}{2} < \frac{\alpha - \beta}{2} < \frac{3\pi}{2}$. This range covers the second and third quadrants, where the cosine function is strictly negative. Ignoring this constraint is the fastest way to lose marks.

Q: Can I use expansion formulas like $\sin(\alpha - \beta)$ instead?

You could try, but it would lead to a system of equations that is significantly more complex to solve. The sum-to-product identities are specifically designed for this 'sum of trig functions' scenario. Always look for the path of least resistance—in JEE, the most elegant method is usually the intended one.

Question

Let $\alpha, \beta$ be such that $\pi < \alpha - \beta < 3\pi$. If $\sin \alpha + \sin \beta = -21/65$ and $\cos \alpha + \cos \beta = -27/65$, then the value of $\cos \frac{\alpha - \beta}{2}$ is

Options:

(A) $-6/65$

(B) $3/\sqrt{130}$

(C) $6/65$

(D) $-3/\sqrt{130}$

Sigma Percentile · Accepted Answer

### Analyzing the Setup

When you first look at the system $\sin \alpha + \sin \beta = -\frac{21}{65}$ and $\cos \alpha + \cos \beta = -\frac{27}{65}$, it is easy to feel overwhelmed. You might be tempted to reach for expansion formulas or complex substitutions. But stop. Take a breath.

In JEE Advanced, the problem is rarely about brute force; it is about recognizing the structure. We are given the sum of two trigonometric functions, and we need to find the cosine of their half-difference.

This is a classic signal to use the sum-to-product identities. Think of these identities as a bridge. They allow us to cross from the world of 'sums' into the world of 'products.'

By applying the identities:

$$2 \sin \left( \frac{\alpha + \beta}{2} ight) \cos \left( \frac{\alpha - \beta}{2} ight) = -\frac{21}{65}$$

$$2 \cos \left( \frac{\alpha + \beta}{2} ight) \cos \left( \frac{\alpha - \beta}{2} ight) = -\frac{27}{65}$$

Suddenly, the term we are hunting for—$\cos \left( \frac{\alpha - \beta}{2} ight)$—appears in both equations.

### The Elimination Strategy

Now, we have our target, but it is shackled to the term $\frac{\alpha + \beta}{2}$. We need to get rid of this unwanted variable. We look for the most fundamental truth in trigonometry: the Pythagorean identity, $\sin^2 	heta + \cos^2 	heta = 1$.

If we square both of our new equations and add them together, the terms involving $\frac{\alpha + \beta}{2}$ will align perfectly. When we perform the operation:

$$\left[ 2 \sin \left( \frac{\alpha + \beta}{2} ight) \cos \left( \frac{\alpha - \beta}{2} ight) ight]^2 + \left[ 2 \cos \left( \frac{\alpha + \beta}{2} ight) \cos \left( \frac{\alpha - \beta}{2} ight) ight]^2 = \left( -\frac{21}{65} ight)^2 + \left( -\frac{27}{65} ight)^2$$

Factoring out the common $4 \cos^2 \left( \frac{\alpha - \beta}{2} ight)$ leaves us with $\sin^2 \left( \frac{\alpha + \beta}{2} ight) + \cos^2 \left( \frac{\alpha + \beta}{2} ight)$ inside the parentheses. Just like that, the complexity vanishes:

$$4 \cos^2 \left( \frac{\alpha - \beta}{2} ight) = \frac{441 + 729}{4225} = \frac{1170}{4225}$$

Simplifying this, we find:

$$\cos^2 \left( \frac{\alpha - \beta}{2} ight) = \frac{1170}{4 	imes 4225} = \frac{1170}{16900} = \frac{9}{130}$$

### The Final Hurdle: The Quadrant Trap

We have calculated the square of our target, $\cos^2 \left( \frac{\alpha - \beta}{2} ight) = \frac{9}{130}$. Taking the square root gives us $\pm \frac{3}{\sqrt{130}}$.

But here is where the JEE examiner tests your attention to detail. We are given the constraint $\pi < \alpha - \beta < 3\pi$. If we divide this inequality by 2, we find that our angle $\frac{\alpha - \beta}{2}$ must lie between $\frac{\pi}{2}$ and $\frac{3\pi}{2}$.

On the unit circle, this corresponds to the second and third quadrants. In these regions, the x-coordinate—which represents the cosine value—is strictly negative.

Therefore, we must reject the positive root. The elegance of this problem lies not just in the algebra, but in the awareness of the domain.

The final answer is **$-\frac{3}{\sqrt{130}}$**.

Animated Solution for Mathematics - Trigonometry: Let $α, β$ be such that $π < α - β < 3 π$ . If $sin α + sin β = - 21/65$ and $cos α + cos β = - 27/65$ , then the value of $cos \frac{α - β}{2}$ is

Select Answer:

Visualized Solution

The Given System

Sum-to-Product Identities

Applying the Identities

The Elimination Strategy

Squaring and Adding

Factoring the Common Term

Applying the Pythagorean Identity

Simplifying the Right Side

Reducing the Fraction

Taking the Square Root

Analyzing the Constraint

Identifying the Quadrants

Final Answer

The Sigma Insight: Trigonometric Ratios and Identities

Analyzing the Setup

The Elimination Strategy

The Final Hurdle

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Animated Solution for Mathematics - Trigonometry: .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Let α,β be such that π<α−β<3π. If sinα+sinβ=−21/65 and cosα+cosβ=−27/65, then the value of cos2α−β​ is

Select Answer:

Visualized Solution

The Given System

Sum-to-Product Identities

Applying the Identities

The Elimination Strategy

Squaring and Adding

Factoring the Common Term

Applying the Pythagorean Identity

Simplifying the Right Side

Reducing the Fraction

Taking the Square Root

Analyzing the Constraint

Identifying the Quadrants

Final Answer

The Sigma Insight: Trigonometric Ratios and Identities

Analyzing the Setup

The Elimination Strategy

The Final Hurdle

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Animated Solution for Mathematics - Trigonometry: Let $α, β$ be such that $π < α - β < 3 π$ . If $sin α + sin β = - 21/65$ and $cos α + cos β = - 27/65$ , then the value of $cos \frac{α - β}{2}$ is

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