TargetIIT

Find range of y given that y=2tan[p]-1[/p]x+sin[p]-1[/p][frac]2x[/]1+x[p]2[/p][/frac]

hint: what is 2 tan [p]-1[/p]x ??

2tan[p]-1[/p]  x = tan[p]-1[/p] {2x / (1-x[p]2[/p])}.....if -1≤x≤1
                     =  tan[p]-1[/p] {2x / (1-x[p]2[/p])} + π.......if x>1
                     =  tan[p]-1[/p] {2x / (1-x[p]2[/p])} -π......if x<-1

also remember that sin2θ = 2 tan θ/(1+tan[p]2[/p] θ)

[hide]Going by those formulae, put x = tanθ such that θ = tan[p]-1[/p]x.
So y = 2θ + sin[p]-1[/p]([frac]2tanθ[/]1+tan[p]2[/p]θ[/frac])
=> y = 2θ + sin[p]-1[/p]sin2θ
=> y = 2θ + 2θ
=> y = 4θ
=> y = 4tan[p]-1[/p]x

Now 0 < [frac]y[/]4[/frac] < [frac]pi[/]2[/frac], as is the principal domain for tan[p]-1[/p]x.

Thus 0 < y < 2pi is the answer.[/hide]

Answer  (-[frac]3π[/]2[/frac],[frac]3π[/]2[/frac])

Shouldnt it be [im]http://codecogs.izyba.com/gif.latex?%5Cleft%28-%5Cfrac%7B3%20%5Cpi%7D%7B4%7D%2C%20%5Cfrac%7B3%20%5Cpi%7D%7B4%7D%20%5Cright%29[/im]?

If you put [im]http://codecogs.izyba.com/gif.latex?x%20%3D%20%5Ctan%20%5Ctheta[/im], where [im]http://codecogs.izyba.com/gif.latex?-%5Cfrac%7B%5Cpi%7D%7B2%7D%20%5Cle%20%5Ctheta%20%5Cle%20%5Cfrac%7B%5Cpi%7D%7B2%7D[/im]then y is given by  

[im]http://codecogs.izyba.com/gif.latex?%5Cbegin%7Bcases%7D%20%26%20-%28%5Cpi%20+%20%5Ctheta%29%20%5Ctext%7B%20if%20%7D%20-%5Cfrac%7B%5Cpi%7D%7B2%7D%20%5Cle%20%5Ctheta%20%5Cle%20-%5Cfrac%7B%5Cpi%7D%7B4%7D%20%5C%5C%20%26%203%20%5Ctheta%20%5Ctext%7B%20if%20%7D%20-%5Cfrac%7B%5Cpi%7D%7B4%7D%20%5Cle%20%5Ctheta%20%5Cle%20%5Cfrac%7B%5Cpi%7D%7B2%7D%20%5C%5C%20%26%20%5Cpi%20-%20%5Ctheta%20%5Ctext%7B%20if%20%7D%20%5Cfrac%7B%5Cpi%7D%7B4%7D%20%5Cle%20%5Ctheta%20%5Cle%20%5Cfrac%7B%5Cpi%7D%7B2%7D%20%5Cend%7Bcases%7D[/im]

Maybe you are right
Thank you :-)