Use the principle of mathematical induction to prove that for all `n in N`
`sqrt(2+sqrt(2+sqrt(2+...+...+sqrt2)))=2cos ((pi)/(2^(n+1)))`
when the LHS contains `n` radical signs.

Explanation

Let P(n) =`sqrt(2+sqrt(2+sqrt2+...+...+sqrt2))=2cos ((pi)/(2^(n+1))).....(i)`
Step-1 For n=1
`"LHS of "(i)=sqrt2 and "RHS of (i) "=2cos((pi)/(2^(2)))`
`2cos((pi)/(4))`
`2.(1)/(sqrt2)`
`=sqrt2` Therefore, P(1) is true.
Step II. Assume it is true for n=k,
`P(k)=underset("k radical sign")(sqrt(2+sqrt(2+sqrt2+...+...+sqrt2)))=2cos ((pi)/(2^(n+1)))`
`=sqrt({2+P(k)})`
`sqrt(2+2cos((pi)/(2^(k+1))))" "("By assumption step")`
`sqrt(2(1+cos((pi)/(2^(k+1)))-1))`
`sqrt(4cos((pi)/(2^(k+2))))`
`2cos((pi)/(2^(k+2)))`
This shows that the result is true for n=k+1. Hence by the principle of mathematical, induction, the result is true for all `n in N`.