If cos A + cos2 A = 1, then sin2 A + sin4 A =

Question:

If $\cos A+\cos ^{2} A=1$, then $\sin ^{2} A+\sin ^{4} A=$

(a) −1
(b) 0
(c) 1
(d) None of these

Solution:

Given:

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

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

So,

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

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

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

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

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

 

$=1$

Hence, the correct option is (c).

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