Question : Find the value of the following expression.
$12\left(\sin^4 \theta+\cos^4 \theta\right)+18\left(\sin^6 \theta+\cos^6 \theta\right)+78 \sin^2 \theta \cos^2 \theta$
Option 1: 30
Option 2: 40
Option 3: 10
Option 4: 20
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Correct Answer: 30
Solution :
$12(\sin^4 \theta+\cos^4 \theta)+18(\sin^6 \theta+\cos^6 \theta)+78 \sin^2 \theta \cos^2 \theta$
$=12[(\sin^2 \theta+\cos^2\theta)^2-2\sin^2 \theta \cos^2 \theta]+18[(\sin^2 \theta+\cos^2 \theta)^3-3\sin^2 \theta \cos^2 \theta(\sin^2 \theta+\cos^2\theta)]+78 \sin^2 \theta \cos^2 \theta$
$=12[1-2\sin^2 \theta \cos^2 \theta]+18[1-3\sin^2 \theta \cos^2 \theta]+78 \sin^2 \theta \cos^2 \theta$
$= 12-24\sin^2 \theta \cos^2 \theta+18-54\sin^2 \theta \cos^2 \theta+78 \sin^2 \theta \cos^2 \theta$
$= 12+18$
$=30$
Hence, the correct answer is 30.
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