Question : Simplify the given expression.
$\frac{1+\sin^4 \theta+\cos^4 \theta}{\cos^2 \theta+\sin^4 \theta}$
Option 1: 3
Option 2: 1
Option 3: 2
Option 4: 4
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Correct Answer: 2
Solution :
Given: $\frac{1+\sin^4 \theta+\cos^4 \theta}{\cos^2 \theta+\sin^4 \theta}$
We know that $\sin^2\theta+\cos^2\theta=1$
Now, $\frac{1+\sin^4 \theta+\cos^4 \theta}{\cos^2 \theta+\sin^4 \theta}$
$=\frac{\sin^2 \theta+\cos^2 \theta+(\sin^2 \theta+\cos^2 \theta)^2–2\sin^2\theta\cos^2\theta}{1–\sin^2 \theta+\sin ^4 \theta}$
$=\frac{1+1–2\sin^2\theta\cos^2\theta}{(\sin^2 \theta)(\sin^2\theta–1)+1}$
$=\frac{2×(1–\sin^2\theta\cos^2\theta)}{1–\sin^2\theta\cos^2\theta}$
$=2$
Hence, the correct answer is 2.
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