Evaluate the following:cosec-1(cosec \(\frac{6\pi}5\))

1.	Evaluate the following:cosec-1(cosec \(\frac{6\pi}5\))
Answer» cosec \((\frac {6\pi}5)\) can be written as cosec (π + \(\frac{\pi}5\)) cosec (π + \(\frac{\pi}5\)) = - cosec\((\frac \pi 5)\) Also, –cosec(θ) = cosec(–θ) ⇒ –cosec\((\frac{\pi}5)\)= cosec\((\frac{-\pi}5)\) Now the question becomes cosec^-1(cosec\((\frac{-\pi}5)\)) cosec^-1(cosec x) = x Provided x ∈ \([\frac{-\pi}2,\frac{\pi}2]\) - {0} ∴ we can write cosec –1(cosec\((\frac{-\pi}5)\)) = \(\frac{-\pi}5.\)

Answer»

cosec \((\frac {6\pi}5)\) can be written as cosec (π + \(\frac{\pi}5\))

cosec (π + \(\frac{\pi}5\)) = - cosec\((\frac \pi 5)\)

Also,

–cosec(θ) = cosec(–θ)

⇒ –cosec\((\frac{\pi}5)\)= cosec\((\frac{-\pi}5)\)

Now the question becomes cosec^-1(cosec\((\frac{-\pi}5)\))

cosec^-1(cosec x) = x

Provided x ∈ \([\frac{-\pi}2,\frac{\pi}2]\) - {0}

∴ we can write cosec –1(cosec\((\frac{-\pi}5)\)) = \(\frac{-\pi}5.\)

Discussion