1.

If cosecx =-2 and pie

Answer»

\pi\ \textless\ x\ \textless\ \dfrac {3\pi}{2} MEANS x is NOTHING but in THIRD QUADRANT. So,

\quad

\csc x=-2=-\csc\left (\dfrac {\pi}{6}\right)\quad\!\implies\quad\!x=\dfrac {7\pi}{6}

\quad

Then,

\quad

\sin(2x)=\sin\left (\dfrac {14\pi}{6}\right)=\sin\left (\dfrac {\pi}{3}\right)=\bf{\dfrac {\sqrt 3}{2}}\\\\\\\sin (3x)=\sin\left (\dfrac {21\pi}{6}\right)=\sin\left (\dfrac {3\pi}{2}\right)=\bf{-1}\\\\\\\tan(2x)=\tan\left (\dfrac {14\pi}{6}\right)=\tan\left (\dfrac {\pi}{3}\right)=\bf{\sqrt 3}\\\\\\\sin (4x)=\sin\left (\dfrac {28\pi}{6}\right)=\sin\left (\dfrac {2\pi}{3}\right)=\bf{\dfrac {\sqrt 3}{2}}


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