In a triangle ABC, if cot A, cot B, cot C are in A.P., then a², b², c², are in .................. pr

Q: Why can't we just use the Sine Rule immediately on the cotangents?

The Sine Rule relates sides to sines, not cotangents. If you try to jump straight to the Sine Rule, you will get stuck with terms like $\frac{\cos A}{\sin A}$, which doesn't simplify easily into side lengths. We must bridge the gap by converting to sine and cosine first, which allows us to use the sine addition formula.

Q: Is there a faster way to solve this?

While there might be clever geometric shortcuts, the method of converting to sine and cosine is the most robust and reliable approach for JEE Advanced. It builds the necessary intuition for handling trigonometric identities in triangle problems, which is a highly tested skill.

Q: Why does $a^2 + c^2 = 2b^2$ imply that $a^2, b^2, c^2$ are in A.P.?

By definition, three terms $x, y, z$ are in Arithmetic Progression if the common difference is constant, meaning $y - x = z - y$, which simplifies to $2y = x + z$. Here, if $a^2 + c^2 = 2b^2$, then $b^2 - a^2 = c^2 - b^2$. This confirms the common difference is the same, proving they are in A.P.

Question

In a triangle $ABC$, if $\cot A, \cot B, \cot C$ are in A.P., then $a^2, b^2, c^2$, are in .................. progression.

Sigma Percentile · Accepted Answer

### The Geometry of Progressions: A Journey Through Triangles

Welcome, future engineers! Today, we are going to unravel a beautiful problem that sits at the intersection of trigonometry and algebra.

Often, when we see a triangle problem, our first instinct is to draw it. But in JEE Advanced, the real magic happens when we translate geometric properties into the language of equations.

We are given a triangle $ABC$ where the cotangents of the angles are in an Arithmetic Progression (A.P.). Our mission is to discover the progression of the squares of the sides: $a^2, b^2, c^2$.

### Phase 1: The Cotangent Condition

We start with the given condition: $\cot A, \cot B, \cot C$ are in A.P. This means the difference between consecutive terms is constant.

Mathematically, this gives us the elegant relation: 
$$2\cot B = \cot A + \cot C$$

This is our anchor. It is the bridge between the angles of the triangle and the algebraic structure we are looking for.

### Phase 2: The Trigonometric Bridge

To make progress, we need to break down these cotangents using the identity $\cot 	heta = \frac{\cos 	heta}{\sin 	heta}$. Substituting this into our equation, we get:

$$2\left(\frac{\cos B}{\sin B}ight) = \frac{\cos A}{\sin A} + \frac{\cos C}{\sin C}$$

To combine the right-hand side, we find a common denominator, $\sin A \sin C$. The numerator becomes $\sin C \cos A + \cos C \sin A$.

Look closely at this numerator—it is the classic sine addition formula, $\sin(A+C)$. Our equation now reads:

$$\frac{2\cos B}{\sin B} = \frac{\sin(A+C)}{\sin A \sin C}$$

### Phase 3: The Power of the Triangle Property

Here is where the geometry saves the day. In any triangle, $A+B+C = \pi$, which implies $A+C = \pi - B$.

If we take the sine of both sides, we get $\sin(A+C) = \sin(\pi - B) = \sin B$. Substituting this back, our equation simplifies dramatically:

$$\frac{2\cos B}{\sin B} = \frac{\sin B}{\sin A \sin C}$$

Cross-multiplying gives us $2\cos B \sin A \sin C = \sin^2 B$. We have successfully eliminated the fractions and are now ready to connect this to the side lengths.

### Phase 4: The Final Revelation

Now, we invoke the heavy artillery: the Sine Rule and the Cosine Rule. From the Sine Rule, we know $\sin A = \frac{a}{2R}$, $\sin B = \frac{b}{2R}$, and $\sin C = \frac{c}{2R}$.

From the Cosine Rule, we have $\cos B = \frac{a^2 + c^2 - b^2}{2ac}$. Substituting these into our equation, we get:

$$2\left(\frac{a^2 + c^2 - b^2}{2ac}ight) \left(\frac{a}{2R}ight) \left(\frac{c}{2R}ight) = \left(\frac{b}{2R}ight)^2$$

Watch as the terms cancel out with satisfying precision: the $ac$ terms vanish, the $2$s cancel, and the $4R^2$ denominators disappear from both sides.

We are left with $a^2 + c^2 - b^2 = b^2$, which rearranges to $a^2 + c^2 = 2b^2$. This is the definitive condition for **$a^2, b^2, c^2$ to be in an Arithmetic Progression**.

We have arrived at the truth: the progression is **arithmetic**. Keep practicing, keep visualizing, and remember that every complex problem is just a series of simple, beautiful steps waiting to be taken.

Animated Solution for Mathematics - Trigonometry: In a triangle $A B C$ , if $cot A, cot B, cot C$ are in A.P., then $a^{2}, b^{2}, c^{2}$ , are in .................. progression.

Visualized Solution

Visualizing the Triangle and Given Condition

The Arithmetic Progression Condition

Converting Cotangent to Sine and Cosine

Combining the Right-Hand Side Fractions

Applying the Sine Addition Formula

Using the Angle Sum Property of a Triangle

Cross-Multiplying to Simplify

Applying the Cosine Rule

Applying the Sine Rule

Final Algebraic Simplification

Conclusion

The Sigma Insight: Properties of Triangles

The Geometry of Progressions

Phase 1

Phase 2

Phase 3

Phase 4

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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; } In a triangle ABC, if cotA,cotB,cotC are in A.P., then a2,b2,c2, are in .................. progression.