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.

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