Simplify: \(\frac{2\sin^2A-\sin^4A+\cos^4A}{3\sin^4A-2\sin^6A+3\cos^4A-2\cos^6A}\)
1
Numerator: \( \cos^4A-\sin^4A=(\cos^2A-\sin^2A)(\cos^2A+\sin^2A)=\cos^2A-\sin^2A \).
So numerator = \( 2\sin^2A+\cos^2A-\sin^2A=\sin^2A+\cos^2A=1 \).
Use \( \sin^4A+\cos^4A=1-2\sin^2A\cos^2A \) and \( \sin^6A+\cos^6A=1-3\sin^2A\cos^2A \).
Denominator = \( 3(1-2\sin^2A\cos^2A)-2(1-3\sin^2A\cos^2A)=3-6\sin^2A\cos^2A-2+6\sin^2A\cos^2A=1 \).
Expression = \( \frac{1}{1}=1 \).
Hence, the simplified value is 1.
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