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Question
Find the principal value of the following:
`cosec^-1(2cos (2pi)/3)`
Solution
Let `cosec^-1(2cos (2pi)/3)=y`
Then,
`cosec y=2cos (2pi)/3`
We know that the range of the principal value branch is `[-pi/2,pi/2]-{0}.`
Thus,
`cosec y =2cos (2pi)/3=2xx(-1)/2=-1=cosec(-pi/2).`
`=>y=-pi/2in[-pi/2,pi/2],y!=0`
Hence, the principal value of `cosec^-1(2cos (2pi)/3) is -pi/2.`
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