Find the maximum and minimum values of `(sin^(-1)x)^3+(cos^(-1)x)^3,`where `-1lt=xlt=1.`

Find the maximum and minimum values of `(sin^(-1)x)^3+(cos^(-1)x)^3,`w

1.	Find the maximum and minimum values of `(sin^(-1)x)^3+(cos^(-1)x)^3,`where `-1lt=xlt=1.`
Answer» `(sin^-1 x)^3 + (cos^-1 x)^3 = (sin^-1 x + cos^-1 x)^3 - 3sin^-1 x cos^-1 x (sin^-1x + cos^-1 x)` `y = (pi/2)^3 - 3sin^-1 x (pi/2 - sin^-1 x )*(pi/2)` `y= pi^3/8 - 3pi^2/4 sin^-1 x + 3 pi/2 (sin^-1 x)^2` `3 pi/2 theta^2 - 3 pi^2/4 theta + pi^3/8 -y=0 ` `(theta - pi/4)^2 - pi^2/16 + pi^2/12 -2y/(3y) = 0` `(theta- pi/4)^2 + pi^2/48 - (2y)/(3 pi) = 0` `-pi/2 <= sin^-1 x <= pi/2` `-(3pi)/4 <= sin^-1 x - pi/4 <= pi/4` `0 <= (sin^-1 x - pi/4)^2 <= (9pi^2)/(16)` `pi^2/48 <= 24/(3 pi) <= (9pi^2)/16 + pi^2/48` `y max= (7pi^2)/8 ; y min = pi^3/32` Answer

Prove that:`sin^(-1)(12)/(13)+cos^(-1)4/5+tan^(-1)(63)/(16)=pi`
What is greater, `tan1 or tan^(-1)1?`
In a ` A B C ,`, if C is a right angle, then `tan^(-1)(a/(b+c))+tan^(-1)(b/(c+a))=``pi/3`(b) `pi/4`(c) `(5pi)/2`(d) `pi/6`
Prove that:`(i)tan^(-1){(sqrt(1+cosx)+sqrt(1-cosx))/(sqrt(1+cosx)-sqrt(1-cosx))}=pi/4+x/2`,
Prove that `tan(cot^(-1)x)=cot(tan^(-1)x)`
Write the value of `sin^-1(sin1550^@)`
Prove that:`tan^(-1)((1-x^2)/(2x))+cot^(-1)((1-x^2)/(2x))=pi/2`
Write the value of `cos^2(1/2cos^(-1)3/5)`
Solve the equation: `cos^(-1)(a/x)-cos^(-1)(b/x)=cos^(-1)(1/b)-cos^(-1)(1/a)`
If `tan^(-1)x+tan^(-1)y=pi/4,`then write the value of `x+y+x ydot`