Question

$t_1=\left(\sin ^{-1} x\right)^{\sin ^{-1} x} ; t_2=\left(\sin ^{-1} x\right)^{\cos ^{-1} x} ; t_3=\left(\cos ^{-1} x\right)^{\sin ^{-1} x} ; t_4=\left(\cos ^{-1} x\right)^{\cos ^{-1} x}$. Where sgn(x) represents signum function. Then which of the following statements is/are correct.

• A. If $x \in(0, \cos 1)$ then $\operatorname{sgn}\left(t_4-t_2+t_3-t_1\right)=1$
• B. If $x \in\left(\cos 1, \frac{1}{\sqrt{2}}\right)$ then $\log _{10}\left(t_3-t_1+t_4-t_2\right)$ is defined
• C. $x \in(\sin 1,1)$ then $t_1 >t_2 >t_4 >t_3$
• D. $x \in\left(\frac{1}{\sqrt{2}}, \sin 1\right)$ then $t_2 >t_1 >t_3 >t_4$

Answer 1

Answer: (A), (B), (C)

$ x \in(0, \cos 1) \Rightarrow \cos ^{-1} x>1>\sin ^{-1} x $
$ x \in\left(\cos 1, \frac{1}{\sqrt{2}}\right) \Rightarrow \sin ^{-1} x<\cos ^{-1} x<1$
$ x \in(\sin 1,1) \Rightarrow \sin ^{-1} x>1>\cos ^{-1} x$
$ x \in\left(\frac{1}{\sqrt{2}}, \sin 1\right) \Rightarrow \cos ^{-1} x<\sin ^{-1} x<1$