Question : If $\cos A+\cos^2 A=1$, then the value of $\sin^4 A+\sin^6 A$ is:
Option 1: $2$
Option 2: $\cos \mathrm{A}$
Option 3: $1$
Option 4: $\sin A$
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Correct Answer: $\cos \mathrm{A}$
Solution :
Given: $\cos A+\cos^2 A=1$
⇒ $\cos A=1 - \cos^2 A$
⇒ $\cos A= \sin^2 A$
Squaring both sides,
⇒ $\cos^2 A= \sin^4 A$
⇒ $1-\sin^2 A= \sin^4 A$
Multiplying both sides by $\sin^2 A$,
$\sin^2 A-\sin^4 A=\sin^6 A$
⇒ $\sin^2 A=\sin^6 A+\sin^4 A$
⇒ $\sin^6 A+\sin^4 A=1-\cos^2 A$
⇒ $\sin^6 A+\sin^4 A=\cos A$
Hence, the correct answer is $\cos A$.
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