To find t if u(v(t))=1 , u(v)=cosv and v(t)=t^2.
Given u(v) = cosv and v(t) = t^2.
Therefore the composite function u(v(t)) = cost^2.
Therefore u(v(t)) = 1 implies cost^2 = 1.
cost^2 = 1 => t^2 = 2n*pi, where n = 0,1,2,3,.....
t = +sqrt(2n*pi), or t = -sqrt(2n*pi), for n = 0,1,2,3...
To evaluate the value of t,we'll have to determine first the composition of the given functions u and v.
u(v(t)) is the result of composing u and v:
(uov)(t) = u(v(t))
To determine the expression of the composed function, we'll substitute v by v(t)) and we'll get:
u(v(t)) = cos v(t)
Now, we'll substitute v(t) by it's expression:
u(v(t)) = cos t^2
But, from enunciation, u(v(t))= 1, so:
cos t^2 = 1
t^2 = +/-arccos 1 + 2*k*pi
t^2 = 0 + 2*k*pi
t = +/-sqrt 2kpi
t = 0