If sina + sin^2a = 1, then what will be cos^2a + cos^4a ?

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164
dk_ch Share
Dec 14, 2016

Given
$\sin a + {\sin}^{2} a = 1$

$\implies \sin a = 1 - {\sin}^{2} a$

$\implies \sin a = {\cos}^{2} a$

$\implies {\sin}^{2} a = {\cos}^{4} a$

$\implies 1 - {\cos}^{2} a = {\cos}^{4} a$

$\implies {\cos}^{2} a + {\cos}^{4} a = 1$

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Explanation

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66
Noah G Share
Dec 14, 2016

$1$

Explanation:

Our starting goal is to turn all terms into cosine. Use the identity ${\sin}^{2} \theta + {\cos}^{2} \theta = 1$.

$\sin a + 1 - {\cos}^{2} a = 1$

$\sin a - {\cos}^{2} a = 0$

$\sin a = {\cos}^{2} a$

Square both sides to get rid of the sine.

${\left(\sin a\right)}^{2} = {\left({\cos}^{2} a\right)}^{2}$

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

Reuse ${\sin}^{2} \theta + {\cos}^{2} \theta = 1$:

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

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

Hopefully this helps!

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