Question

6. If (, ) = sin−1 ( √−√ − ) , 0 < < 1, then prove that = − .

Answer 1

Answer:

Solution

E=sin−1x1−−−−−−√cos−1x2−−−−−−−√+sin−1x2−−−−−−√cos−1x3−−−−−−−√+sin−1x3−−−−−−√cos−1x4−−−−−−−√+...+sin−1x28−−−−−−−√cos−1x1−−−−−−−√

xi∈[0,1]∀i=1,2,3,..,28

∴sin−1xi>0

Now using A.M.≥G.M., we have

a2+b22≥ab, where a,b>0

∴sin−1x1−−−−−−√cos−1x2−−−−−−−√≤(sin−1x1+cos−1x22)

sin−1x2−−−−−−√cos−1x3−−−−−−−√<(sin−1x2+cos−1x32)⋮ ⋮sin−1x28−−−−−−−√cos−1x1−−−−−−−√≤(sin−1x28+cos−1x12)

On adding all, we get

E≤∑28i=1sin−1xi+cos−1xi2

∴E≤28(π2)2

∴Emax=7π