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 $.

AcademicMathematicsNCERTClass 10

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.

raja
Updated on 10-Oct-2022 13:28:58

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