Question

Question #3dacd

Answer 1

Let

`y=(sin^-1x)^2+(cos^-1x)^2`

`=>y=(sin^-1x+cos^-1x)^2-2*sin^-1x*cos^-1x`

Now putting `cos^-1x=pi/2-sin^-1x` we get

`y=(sin^-1x + pi/2-sin^-1x)^2-2*sin^-1x*(pi/2-sin^-1x)`

`=>y=(pi/2)^2-2*sin^-1x*(pi/2-sin^-1x)`

We know that range of `sin^-1x in [-pi/2,pi/2]`

So we get greatest value of `y` for `sin^-1x=-pi/2`

`y_"greatest"=pi^2/4-2(-pi/2)(pi/2-(-pi/2))`
`=pi^2/4+pi^2=5/4pi^2`

And we get least value of `y` for `sin^-1x=pi/2`

`y_"least"=pi^2/4-2(pi/2)(pi/2-pi/2)`

`=pi^2/4+0=pi^2/4`

This is evident from the graph given below

graph{(arcsinx)^2+(arccosx)^2 [-5.173, 5.927, -1.98, 3.57]}