If $5\theta$ and $4\theta$ are acute angles satisfying $\sin5\theta=\cos4\theta,$ then $2\sin3\theta-\sqrt{3}\tan3\theta$ is equal to:
A$1$
B$0$
C$-1$
D$1+\sqrt{3}$
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B$0$
We are given that $5\theta$ and $4\theta$ are acute angles satisfying the following condition
$\sin5\theta=\cos4\theta$
We are asked to find $2\sin3\theta-\sqrt{3}\tan3\theta$
$\Rightarrow\sin5\theta=\cos4\theta$
$\Rightarrow\cos(90^\circ-5\theta)=\cos4\theta$
$\Rightarrow90^\circ-5\theta=4\theta$
$\Rightarrow9\theta=90^\theta$
Where $5\theta$ and $4\theta$ are acute angles
$\Rightarrow\theta=10^\circ$
Now we have to find:
$2\sin3\theta-\sqrt{3}\tan3\theta$
$=2\sin30^\circ-\sqrt{3}\tan30^\circ$
$=2\times\frac{1}{2}-\sqrt{3}\times\frac{1}{\sqrt{3}}$
$=1-1$
$=0$
Hence the correct option is $(b)$
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