If 2 x 3 4 , then cos – 1 ( 12 13 cos x + 5 13 sin x ) is equal to [2025]

Question

If $\frac{π}{2} \leq x \leq \frac{3 π}{4}$ , then $\cos^{- 1} (\frac{12}{13} \cos x + \frac{5}{13} \sin x)$ is equal to [2025]

Smart Achievers · Accepted Answer

(1)

Given, $\frac{π}{2} \leq x \leq \frac{3 π}{4}$

Let $\cos α = \frac{12}{13} [∴ \cos^{- 1} (\frac{12}{13} \cos x + \frac{5}{13} \sin x)]$

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

$= x - \tan^{- 1} (\frac{5}{12})$

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