(i) Given as cosec^-1 (-√2)

Let y = cosec^-1 (-√2)

cosec y = -√2

– cosec y = √2

– cosec (π/4) = √2

– cosec (π/4) = cosec (-π/4) [since – cosec θ = cosec (-θ)]

So, the range of principal value of cosec^-1 [-π/2, π/2] – {0} and cosec (-π/4) = – √2

cosec (-π/4) = – √2

So, the principal value of cosec^-1 (-√2) is – π/4

(ii) Given as cosec^-1 (-2)

Let y = cosec^-1 (-2)

cosec y = -2

– cosec y = 2

– cosec (π/6) = 2

– cosec (π/6) = cosec (-π/6) [since – cosec θ = cosec (-θ)]

So, the range of principal value of cosec^-1 [-π/2, π/2] – {0} and cosec (-π/6) = – 2

cosec (-π/6) = – 2

So, the principal value of cosec^-1 (-2) is – π/6

(iii) Given as cosec^-1 (2/√3)

Let y = cosec^-1 (2/√3)

cosec y = (2/√3)

cosec (π/3) = (2/√3)

So, the range of principal value of cosec^-1 is [-π/2, π/2] – {0} and cosec (π/3) = (2/√3)

Hence, the principal value of cosec^-1 (2/√3) is π/3

(iv) Given as cosec^-1 (2 cos(2π/3))

As we know that cos (2π/3) = – ½

So, 2 cos (2π/3) = 2 × – ½

2 cos (2π/3) = -1

Substitute these values in cosec^-1 (2 cos(2π/3)) we get,

cosec^-1 (-1)

Let y = cosec^-1 (-1)

– cosec y = 1

– cosec (π/2) = cosec (-π/2) [since –cosec θ = cosec (-θ)]

So, the range of principal value of cosec^-1 [-π/2, π/2] – {0} and cosec (-π/2) = – 1

cosec (-π/2) = – 1

So, the principal value of cosec^-1 (2 cos(2π/3)) is – π/2