Question

What is sec^-1⁡x in terms of tan^-1⁡?

(a) tan^-1⁡\(\sqrt{1+x^2}\)

(b) tan^-1⁡1+x^2

(c) tan^-1⁡x

(d) tan^-1⁡\(\sqrt{x^2-1}\)

This question was posed to me by my college professor while I was bunking the class.

My question comes from Properties of Inverse Trigonometric Functions topic in section Inverse Trigonometric Functions of Mathematics – Class 12

Answer 1

Right option is (d) tan^-1⁡\(\sqrt{x^2-1}\)

For explanation: Let sec^-1⁡x=y

⇒x=sec⁡y

⇒x=\(\sqrt{1+tan^2⁡y}\)

⇒x^2-1=tan^2⁡y

∴y=tan^-1⁡\(\sqrt{x^2-1}\)=sec^-1⁡x.