2010/008) Show that there is exactly one value of x which satisfies the equation,?

2 cos^2 (x^3 + x) = 2^x + 2^(-x).

proof:
the maximum value of LHS = 2

as cos^2(x^3+x) <= 1

the minimum value of RHS

let 2^x = y

so y + 1/y = (sqrt(y) - 1/(sqrt(y))^2 + 2

so minum value of RHS = 2


so both sides are same when both are 2

RHS = 2 when y = 1 or x= 0

LHS = 2 when 2 cos^2(x^3+x) = 2

or cos^2(x^3+x) = 1

x = 0 satisies LHS

so x = 0 is the only value that satisfied equality