Q. If $\frac\pi2\le x\le\frac{3\pi}4$, then $\cos^{-1}\left(\frac{12}{13}\cos x+\frac5{13}\sin x\right)$ is equal to
A) $x-\tan^{-1}\frac43$
B) $x+\tan^{-1}\frac45$
C) $x-\tan^{-1}\frac5{12}$
D) $x+\tan^{-1}\frac5{12}$
Explanation
[Solution by Dr. Abubacker] Let $\cos\varphi=\frac{12}{13}$, $\sin\varphi=\frac5{13}$ ($\varphi=\tan^{-1}\frac5{12}$). The expression is $\cos^{-1}\cos(x-\varphi)$. For $\frac\pi2\le x\le\frac{3\pi}4$, $x-\varphi\in[0,\pi]$, so the value is $x-\tan^{-1}\frac5{12}$. Answer (3).
Reviewed by Dr. Abubacker
Physics Educator & NEET/JEE Specialist
Ph.D. in Physics with 19+ years of teaching experience.