Prove this identity (1 Viewer)

[currysauce]

root((k+1)/2) - root((k-1)/2) = root (k - root(k^2-1))

 

[icycloud]

Let LHS = Sqrt((k+1)/2) - Sqrt((k-1)/2)
LHS² = (k+1)/2 + (k-1)/2 - 2Sqrt((k+1)(k-1)/4)
= k - Sqrt(k^2-1)
Thus, LHS = Sqrt(LHS²) = Sqrt(k-Sqrt(k^2-1)) = RHS

 

[NightShadow]

what about restrictions because you're square rooting? wont you have a negative answer component as well?

 

[currysauce]

thankyou

 

[icycloud]

Well, k >= 1 for LHS to be defined.
And LHS >= 0 for k >= 1.
Therefore, we take the positive root for RHS.