Question : If $3 \sin x+4 \cos x=2$, then the value of $3 \cos x – 4 \sin x$ is equal to:
Option 1: $\sqrt{23}$
Option 2: $\sqrt{21}$
Option 3: $\sqrt{29}$
Option 4: $21$
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Correct Answer: $\sqrt{21}$
Solution : Given: $3 \sin x+4 \cos x=2$ (equation 1) Let $3 \cos x–4 \sin x=k$ (equation 2) We know the trigonometric identity, $\sin^2 x+ \cos^2 x=1$. Squaring the equation and adding them together, $(3 \cos x–4 \sin x)^2+(3 \sin x+4 \cos x)^2=k^2+2^2$ ⇒ $9 \cos^2 x+16 \sin^2 x–24\sin x \cos x+9 \sin^2 x+16 \cos^2 x+24 \sin x \cos x=k^2+4$ ⇒ $9( \cos^2 x+ \sin^2 x)–24\sin x \cos x+16 (\sin^2 x+ \cos^2 x)+24 \sin x \cos x=k^2+4$ ⇒ $9+16=k^2+4$ ⇒ $25=k^2+4$ ⇒ $25–4=k^2$ ⇒ $k^2=21$ ⇒ $k=\sqrt{21}$ The value of $3 \cos x–4 \sin x=\sqrt {21}$. Hence, the correct answer is $\sqrt {21}$.
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