1) The greatest value of (sin-1x ) 3 +( cos-1 x) 3
2) The value of sin(cot-1 (cos(tan-1 x))
3) The smallest and the largest values of tan-1 [( 1-x) / (1+x)]
3(sin-1 x + cos -1 x)((sin-1x)^2 - sin-1xcos-1x + (cos-1x)^2 ...........
(pi/2)(pi^2/4 - 3sin-1xcos-1x) ...........
from here it shud be easy[1]
3say tan@ =x ............
cos@ = 1/√(1+x^2).......
now it is
sin(cot-1(1/√(1+x^2)))...........
say cot# = 1/√(1+x^2).......
sin# = √(1+x^2)/√(1+(1+x^2)^2)............
or the whole exp. is .........
√[(1+x^2)/(1+(1+x^2)^2)]..........
from here i think differentiation will do the job to find maxima!!!!!!
3tan-1[(1-x)/(1+x)] = tan-1(1) - tan-1(x) .........
pi/4 - tan-1(x) ............
-pi/2<tan-1x<pi/2............
so pi/4+ pi/2 is maximum ....
3 pi /4 shud be maxima!!!!!![1]
