Evaluate the following:cosec-1\(\{cosec(-\frac{9π}4)\}\)

1.	Evaluate the following:cosec-1\(\{cosec(-\frac{9π}4)\}\)
Answer» As we know cosec(–θ) = –cosec θ ∴ cosec\((\frac{-9\pi}4)\) = -cosec\((\frac{9\pi}4)\) -cosec\((\frac{9\pi}4)\) can be written as -cosec\((2\pi+\frac{\pi}4)\) Also, cosec (2π+θ) = cosec θ ∴ -cosec\((2\pi+\frac{\pi}4)\) = -cosec\((\frac{\pi}4)\) As we know –cosec (θ) = cosec (–θ) ∴ -cosec\((\frac{\pi}4)\) = cosec\((\frac{-\pi}4)\) Now the question becomes cosec^-1(cosec\((\frac{-\pi}4)\)) cosec^-1(cosec x) = x Provided x ∈ \([\frac{-\pi}2,\frac{\pi}2]\) - {0} ∴ We can write cosec^-1(cosec\((\frac{-\pi}4)\)) = \(\frac{-\pi}4.\)

Answer»

As we know cosec(–θ) = –cosec θ

∴ cosec\((\frac{-9\pi}4)\) = -cosec\((\frac{9\pi}4)\)

-cosec\((\frac{9\pi}4)\) can be written as -cosec\((2\pi+\frac{\pi}4)\)

Also,

cosec (2π+θ) = cosec θ

∴ -cosec\((2\pi+\frac{\pi}4)\) = -cosec\((\frac{\pi}4)\)

As we know –cosec (θ) = cosec (–θ)

∴ -cosec\((\frac{\pi}4)\) = cosec\((\frac{-\pi}4)\)

Now the question becomes cosec^-1(cosec\((\frac{-\pi}4)\))

cosec^-1(cosec x) = x

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

∴ We can write cosec^-1(cosec\((\frac{-\pi}4)\)) = \(\frac{-\pi}4.\)

Discussion