If x=(5)/(12) for some x ∈(π, (3 π)/(2)) , then 7 x( (13 x)/(2)+ (13 x)/(2))+ 7 x( (13 x)/(2)- (13 x)/(2)) is equal to

Correct answer: D. 1 13 x = (5)/(12) x = (-5)/(13) = 2 ^2 (x)/(2) - 1 ((x)/(2)) = - 2 13 or 2 13 (rejected) \ (x)/(2) ∈ ((π)/(2), (3π)/(4)) \ ( 7x ( 13x)/(2) + 7x ( 13x)/(2)) + ( 7x ( 13x)/(2) - 7x ( 13x)/(2)) (7x - (13x)/(2)) + (7x - (13x)/(2)) (x)/(2) + ((x)/(2)) 3 13 - 2 13 = 1 13

JEE Main 202624 JanuaryMorning Shift

Exam: JEE Main
Year: 2026
Date: 24 January
Shift: Morning Shift
Subject: Mathematics
Chapter: Trigonometric Ratios & Identities

JEE Main 2026 · 24 January · Morning Shift · Mathematics · Trigonometric Ratios & Identities

If x=(5)/(12) for some x ∈(π, (3 π)/(2)) , then 7 x( (13 x)/(2)+...

\cot x=\frac{5}{12}

for some

x \in\left(\pi, \frac{3 \pi}{2}\right)

, then

\sin 7 x\left(\cos \frac{13 x}{2}+\sin \frac{13 x}{2}\right)+\cos 7 x\left(\cos \frac{13 x}{2}-\sin \frac{13 x}{2}\right)

is equal to

Options

A $\frac{4}{\sqrt{26}}$
B $\frac{6}{\sqrt{26}}$
C $\frac{5}{\sqrt{13}}$
D $\frac{1}{\sqrt{13}}$

Correct answer

D. $\frac{1}{\sqrt{13}}$

Step-by-step solution

\cot x = \dfrac{5}{12} \Rightarrow \cos x = \dfrac{-5}{13} = 2\cos^2 \dfrac{x}{2} - 1

\cos\left(\dfrac{x}{2}\right) = -\dfrac{2}{\sqrt{13}}

\dfrac{2}{\sqrt{13}}

(rejected)
\left\{ \because \dfrac{x}{2} \in \left(\dfrac{\pi}{2}, \dfrac{3\pi}{4}\right) \right\}

<br>\left(\sin 7x \dfrac{\sin 13x}{2} + \cos 7x \dfrac{\cos 13x}{2}\right) + \left(\sin 7x \dfrac{\cos 13x}{2} - \cos 7x \dfrac{\sin 13x}{2}\right)

\cos\left(7x - \dfrac{13x}{2}\right) + \sin\left(7x - \dfrac{13x}{2}\right)

\cos \dfrac{x}{2} + \sin\left(\dfrac{x}{2}\right)

\dfrac{3}{\sqrt{13}} - \dfrac{2}{\sqrt{13}} = \dfrac{1}{\sqrt{13}}

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