Q. Using the principal values of the inverse trigonometric functions, the sum of the maximum and the minimum values of $16\left((\sec^{-1}x)^2+(\operatorname{cosec}^{-1}x)^2\right)$ is :
A) $24\pi^2$
B) $22\pi^2$
C) $31\pi^2$
D) $18\pi^2$
Explanation
[Solution by Dr. Abubacker] Let $a=\sec^{-1}x\in[0,\pi]\setminus\{\frac\pi2\}$, so $\operatorname{cosec}^{-1}x=\frac\pi2-a$. The expression is $16\left(2a^2-\pi a+\frac{\pi^2}4\right)$: minimum at $a=\frac\pi4$ is $2\pi^2$; maximum at $a=\pi$ is $20\pi^2$. Sum $=22\pi^2$. Answer (2).
Reviewed by Dr. Abubacker
Physics Educator & NEET/JEE Specialist
Ph.D. in Physics with 19+ years of teaching experience.