Question : Find the value of the following expression.
$5\left(\sin ^4 \theta+\cos ^4 \theta\right)+3\left(\sin ^6 \theta+\cos ^6 \theta\right)+19 \sin ^2 \theta \cos ^2 \theta$
Option 1: 8
Option 2: 5
Option 3: 6
Option 4: 7
Correct Answer: 8
Solution :
$5(\sin ^4 \theta+\cos ^4 \theta)+3(\sin ^6 \theta+\cos ^6 \theta)+19 \sin ^2 \theta \cos ^2 \theta$
$=5[(\sin ^2 \theta+\cos ^2 \theta)^2-2\sin ^2 \theta\cos ^2 \theta]+3[(\sin ^2 \theta+\cos ^2 \theta)^3-3\sin ^2 \theta\cos ^2 \theta(\sin ^2 \theta+\cos ^2 \theta)] + 19 \sin ^2 \theta \cos ^2 \theta$
$= 5[(1)^2-2\sin ^2 \theta\cos ^2 \theta]+3[(1)^3-3\sin ^2 \theta\cos ^2 \theta(1)] + 19 \sin ^2 \theta \cos ^2 \theta$
$= 5-10\sin ^2 \theta\cos ^2 \theta+3-9\sin ^2 \theta\cos ^2 \theta + 19 \sin ^2 \theta \cos ^2 \theta$
$=8-19\sin ^2 \theta\cos ^2 \theta+19 \sin ^2 \theta \cos ^2 \theta$
= 8
Hence, the correct answer is 8.
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