Mathematics

Problem

Let

x

be any real number such that

0 < x < 1

. a. (i) Explain why both

\arcsin x

and

\arctan x

lie in the interval

\left(0,\tfrac{\pi}{2}\right)

. b. (ii) Prove that

\arcsin x > \arctan x

for every

x

\left(0,1\right)

. c. (iii) Determine whether there exists a value of

x

\left(0,1\right)

for which

\arcsin x = \arctan x

. Justify your answer.