If π/2 leq x leq 3 π/4 , then cos -1 (12/13 cos x+5/13 sin x ) is equal to

Mathematics · Inverse Trigonometric Functions

JEE Main 2025 — 23 January, Morning Shift — Question 14

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

is equal to

$\frac{\pi}{2} \leq \mathrm{x} \leq \frac{3 \pi}{4}$

$\cos ^{-1}\left(\frac{12}{13} \cos x+\frac{5}{13} \sin x\right)$

$\cos ^{-1}(\cos x \cos \alpha+\sin x \sin \alpha)$

$\cos ^{-1}(\cos (x-\alpha))$

$\Rightarrow \mathrm{x}-\alpha$ because $\mathrm{x}-\alpha \in\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)$

$\Rightarrow \mathrm{x}-\tan ^{-1} \frac{5}{12}$ .

Answer key and solution verified before publishing.

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