arcsec(x) + arccsc(x) = \(\frac{π}{2}\)

We will learn how to prove the property of the inverse trigonometric function arcsec(x) + arccsc(x) = \(\frac{π}{2}\) (i.e., sec\(^{-1}\) x + csc\(^{-1}\) x = \(\frac{π}{2}\)).

Proof: Let, sec\(^{-1}\) x = θ              

Therefore, x = sec θ

x = csc (\(\frac{π}{2}\) - θ), [Since, csc (\(\frac{π}{2}\) - θ) = sec θ]

⇒ csc\(^{-1}\) x = \(\frac{π}{2}\) - θ

⇒ csc\(^{-1}\) x= \(\frac{π}{2}\) - sec\(^{-1}\) x, [Since, θ = sec\(^{-1}\) x]

⇒ csc\(^{-1}\) x + sec\(^{-1}\) x = \(\frac{π}{2}\)

⇒ sec\(^{-1}\) x + csc\(^{-1}\) x = \(\frac{π}{2}\)

Therefore, sec\(^{-1}\) x + csc\(^{-1}\) x = \(\frac{π}{2}\).         Proved.

11 and 12 Grade Math

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