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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