Question : If $(4 \sin \theta+5 \cos \theta)=3$, then the value of $(4 \cos \theta-5 \sin \theta)$ is:
Option 1: $3 \sqrt{2}$
Option 2: $4 \sqrt{2}$
Option 3: $2 \sqrt{3}$
Option 4: $2 \sqrt{5}$
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Correct Answer: $4 \sqrt{2}$
Solution :
$(4 \sin \theta+5 \cos \theta)=3$
Squaring both sides, we get,
$⇒16 \sin^2 \theta+25 \cos^2 \theta+40\sin\theta\cos\theta=9$
$⇒16(1-\cos^2 \theta)+25(1-\sin^2 \theta) + 40\sin\theta\cos\theta= 9$
$⇒16-16\cos^2\theta+25-25\sin^2\theta+40\sin\theta\cos\theta=9$
$⇒16\cos^2\theta+25\sin^2\theta-40\sin\theta\cos\theta=16+25-9$
$⇒(4\cos\theta)^2+(5\sin\theta)^2-2\times 5\sin\theta \times 4\cos\theta=32$
$⇒(4\cos\theta - 5\sin\theta)^2 = 32$
$⇒(4\cos\theta - 5\sin\theta) = \sqrt{32}=4\sqrt{2}$
Hence, the correct answer is $4\sqrt{2}$.
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