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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