If sin A + sin2 A = 1,

Here the given date is $\sin A+\sin ^{2} A=1$ and

We have to find the value of $\cos ^{2} A+\cos ^{4} A$

We know that the given relation is

$\sin A+\sin ^{2} A=1 \ldots \ldots(1)$

Now we are going to evaluate the value of

$\cos ^{2} A+\cos ^{4} A$

$=\left(\cos ^{2} A\right)+\left(\cos ^{2} A\right)^{2}$

$=\left(1-\sin ^{2} A\right)+\left(1-\sin ^{2} A\right)^{2}$

$=\sin A+\sin ^{2} A$

Here we are using the relation $\sin ^{2} A+\cos ^{2} A=1$

This is same as the equation number (1)

Therefore $\cos ^{2} A+\cos ^{4} A=1$

Hence the option (b) is correct.