Question : $\cos^4 A-\sin^4 A$ is equal to:
Option 1: $2 \cos^2 A+1$
Option 2: $1-2 \sin ^2 A$
Option 3: $2 \sin^2 A-1$
Option 4: $-\left(2 \sin^2 A+1\right)$
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Correct Answer: $1-2 \sin ^2 A$
Solution :
Given: $\cos^4 A-\sin^4 A$
$= (\cos^2A)^2-(\sin ^2A)^2$
$= (\cos^2A-\sin^2 A)(\cos^2 A+\sin^2 A)$
$= (\cos^2A-\sin^2A)×1$ [$\because \cos^2A+\sin^2A=1$]
$=(1-\sin^2A-\sin^2A)$
$= 1-2\sin^2A$
Hence, the correct answer is $1-2\sin^2A$.
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