Can someone please explain that how we got#1 <=sin^(-1)cos^(-1)sin^(-1)tan^(-1)x<=(pi)/2# from #[sin^(-1)cos^(-1)sin^(-1)tan^(-1)x]=1# ?, where [.] denotes the greatest integer function.

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Answer 1 · 2018-01-19T11:55:01

See below.

For #x in [1,pi/2]# we have #sin(x)# is monotonically increasing then

#sin(1) le cos^-1(sin^-1(tan^-1x)) le 1#

now for #x in [sin(1),sin(pi/2)]# we have that

#cos(x)# is monotonically decreasing then

#cos(sin(1)) ge sin^-1(tan^-1 x) ge cos(1)#

now for #x in [cos(1),cos(sin(1))]# we have

#sin x# is monotonically increasing then

#sin(cos(1)) le tan^-1 x le sin(cos(sin(1)))#

now for #x in [sin(cos(1)),sin(cos(sin(1))) ]# we have that

#tan(x)# is monotonically increasing then

#tan(sin(cos(1))) le x le tan(sin(cos(sin(1))))#

now calling

#x_1 = tan(sin(cos(1)))# and #x_2 = tan(sin(cos(sin(1))))#

we have

#sin^-1(cos^-1(sin^-1(tan^-1 x_1))) = pi/2# and
#sin^-1(cos^-1(sin^-1(tan^-1 x_2))) = 1#