If $ \sin \mathrm{A}+\sin ^{2} \mathrm{~A}=1 $, then the value of the expression $ \left(\cos ^{2} \mathrm{~A}+\cos ^{4} \mathrm{~A}\right) $ is
(A) 1
(B) $ \frac{1}{2} $
(C) 2
(D) 3

Given:

\( \sin \mathrm{A}+\sin ^{2} \mathrm{~A}=1 \)

To do:

We have to find the value of the expression \( \left(\cos ^{2} \mathrm{~A}+\cos ^{4} \mathrm{~A}\right) \).

Solution:  

We know that,

$\sin ^{2} A+\cos ^{2} A=1$

Therefore,

$\sin A+\sin ^{2} A =1$

$\sin A =1-\sin ^{2} A$

$=\cos ^{2} A$

Squaring on both sides, we get,

$\sin ^{2} A=\cos ^{4} A$

$1-\cos ^{2} A =\cos ^{4} A$

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