Write 'True' or 'False' and justify your answer in each of the following:
If $ \cos \mathrm{A}+\cos ^{2} \mathrm{~A}=1 $, then $ \sin ^{2} \mathrm{~A}+\sin ^{4} \mathrm{~A}=1 $.
Given:
If \( \cos \mathrm{A}+\cos ^{2} \mathrm{~A}=1 \), then \( \sin ^{2} \mathrm{~A}+\sin ^{4} \mathrm{~A}=1 \).
To do:
We have to find whether the given statement is true or false.
Solution:
We know that,
$\sin ^{2} A+\cos ^{2} A=1$
$\cos ^{2} A=1-\sin ^{2} A$
Therefore,
$\cos A+\cos ^{2} A=1$
$\cos A=1-\cos ^{2} A$
$\cos A=\sin ^{2} A$
Squaring on both sides, we get,
$\cos ^{2} A=\sin ^{4} A$
$1-\sin ^{2} A=\sin ^{4} A$
$\sin ^{2} A+\sin ^{4} A=1$
The given statement is true.
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