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Let cosec-1 {cosec(-π/4)} = x
=> cosec(-π/4) = cosec x
=> -cosec(π/4) = cosec x
=> -√2 = cosec x
=> x = cosec-1 {-√2}
=> x = -cosec-1 {√2}
=> x = -π/4
So, cosec-1 {cosec(-π/4)} = -π/4