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JEE Main 202523 JanuaryMorning Shift
Exam
JEE Main
Year
2025
Date
23 January
Shift
Morning Shift
Subject
Mathematics
Chapter
Trigonometric Ratios & Identities

JEE Main 2025 · 23 January · Morning Shift · Mathematics · Trigonometric Ratios & Identities

If (π)/(2) ≤ x ≤ (3 π)/(4) , then ^ -1 ((12)/(13) x+(5)/(13) x) is...

If π2≤x≤3π4\frac{\pi}{2} \leq x \leq \frac{3 \pi}{4}, then $\cos ^{-1}\left(\frac{12}{13} \cos x+\frac{5}{13} \sin x\right) is equal to

Options

  1. Ax−tan⁡−143x-\tan ^{-1} \frac{4}{3}
  2. Bx+tan⁡−145x+\tan ^{-1} \frac{4}{5}
  3. Cx−tan⁡−1512x-\tan ^{-1} \frac{5}{12}
  4. Dx+tan⁡−1512x+\tan ^{-1} \frac{5}{12}

Correct answer

C. x−tan⁡−1512x-\tan ^{-1} \frac{5}{12}

Step-by-step solution

1213cos⁡x+513sin⁡x Let tan⁡α=512,α∈(0,π2)⇒sin⁡α=513,cos⁡α=1213\begin{aligned} & \frac{12}{13} \cos x+\frac{5}{13} \sin x \\ & \text{ Let } \tan \alpha=\frac{5}{12}, \alpha \in\left(0, \frac{\pi}{2}\right) \\ & \Rightarrow \sin \alpha=\frac{5}{13}, \cos \alpha=\frac{12}{13}\end{aligned} ⇒1213cos⁡x+513sin⁡x=cos⁡αcos⁡x+sin⁡αsin⁡x=cos⁡(x−α)⇒cos⁡−1[cos⁡(x−α)]=x−α=x−tan⁡−1(512)\begin{aligned} & \Rightarrow \frac{12}{13} \cos x+\frac{5}{13} \sin x=\cos \alpha \cos x+\sin \alpha \sin x \\ & \quad=\cos (x-\alpha) \\ & \Rightarrow \cos ^{-1}[\cos (x-\alpha)]=x-\alpha \\ & \quad=x-\tan ^{-1}\left(\frac{5}{12}\right)\end{aligned}$
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