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JEE Main 20264 AprilMorning Shift
Exam
JEE Main
Year
2026
Date
4 April
Shift
Morning Shift
Subject
Mathematics
Chapter
Trigonometric Ratios & Identities

JEE Main 2026 · 4 April · Morning Shift · Mathematics · Trigonometric Ratios & Identities

If A = ( 3^ )/( 9^ ) + ( 9^ )/( 27^ ) + ( 27^ )/( 81^ ) and B = 81^...

If A=sin⁡3∘cos⁡9∘+sin⁡9∘cos⁡27∘+sin⁡27∘cos⁡81∘A = \dfrac{\sin 3^\circ}{\cos 9^\circ} + \dfrac{\sin 9^\circ}{\cos 27^\circ} + \dfrac{\sin 27^\circ}{\cos 81^\circ} and B=tan⁡81∘−tan⁡3∘B = \tan 81^\circ - \tan 3^\circ, then BA\dfrac{B}{A} is equal to _____.

Correct answer

0

Step-by-step solution

Consider the general term sin⁡xcos⁡3x\dfrac{\sin x}{\cos 3x}.

We can rewrite this by multiplying and dividing by 2cos⁡x2\cos x:

2sin⁡xcos⁡x2cos⁡3xcos⁡x=sin⁡2x2cos⁡3xcos⁡x\dfrac{2\sin x \cos x}{2\cos 3x \cos x} = \dfrac{\sin 2x}{2\cos 3x \cos x}

Using sin⁡2x=sin⁡(3x−x)=sin⁡3xcos⁡x−cos⁡3xsin⁡x\sin 2x = \sin(3x - x) = \sin 3x \cos x - \cos 3x \sin x, we get:

sin⁡3xcos⁡x−cos⁡3xsin⁡x2cos⁡3xcos⁡x=12(sin⁡3xcos⁡3x−sin⁡xcos⁡x)=12(tan⁡3x−tan⁡x)\dfrac{\sin 3x \cos x - \cos 3x \sin x}{2\cos 3x \cos x} = \dfrac{1}{2} \left( \dfrac{\sin 3x}{\cos 3x} - \dfrac{\sin x}{\cos x} \right) = \dfrac{1}{2} (\tan 3x - \tan x)

Applying this identity to each term of AA:

sin⁡3∘cos⁡9∘=12(tan⁡9∘−tan⁡3∘)\dfrac{\sin 3^\circ}{\cos 9^\circ} = \dfrac{1}{2} (\tan 9^\circ - \tan 3^\circ)

sin⁡9∘cos⁡27∘=12(tan⁡27∘−tan⁡9∘)\dfrac{\sin 9^\circ}{\cos 27^\circ} = \dfrac{1}{2} (\tan 27^\circ - \tan 9^\circ)

sin⁡27∘cos⁡81∘=12(tan⁡81∘−tan⁡27∘)\dfrac{\sin 27^\circ}{\cos 81^\circ} = \dfrac{1}{2} (\tan 81^\circ - \tan 27^\circ)

Adding these terms, the series telescopes:

A=12(tan⁡9∘−tan⁡3∘+tan⁡27∘−tan⁡9∘+tan⁡81∘−tan⁡27∘)A = \dfrac{1}{2} (\tan 9^\circ - \tan 3^\circ + \tan 27^\circ - \tan 9^\circ + \tan 81^\circ - \tan 27^\circ)

A=12(tan⁡81∘−tan⁡3∘)A = \dfrac{1}{2} (\tan 81^\circ - \tan 3^\circ)

We are given B=tan⁡81∘−tan⁡3∘B = \tan 81^\circ - \tan 3^\circ.

Substituting BB into the equation for AA:

A=12BA = \dfrac{1}{2} B

⇒BA=2\Rightarrow \dfrac{B}{A} = 2

Answer: 22
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