How do you simplify sec(tan^-1x)?

Inverse trigonometric functions are the inverse ratio of the basic trigonometric ratios.  

Answer: The value of sec(tan^-1x) is √x² + 1.

Let's solve the following.

Explanation:

To Simplify: sec(tan^-1x)

Let, 

y = tan^-1x

⇒ x = tan y

⇒ x = sin y / cos y            (Since, tan y = sin y / cos y)

Squaring on both the sides,

⇒ x² = sin²y / cos²y

Adding 1 on both the sides of the equation,

⇒ x² + 1 = (sin²y / cos²y) + 1

⇒ x² + 1 = (sin²y + cos²y) / cos²y

⇒ x² + 1 = 1 / cos²y               (Since, sin²y + cos²y = 1)

⇒ x² + 1 = sec²y

Taking square root on both the sides,

⇒ √x² + 1 = sec²y

⇒ √x² + 1 = sec(tan^-1x)           (Since, y = tan^-1x)

Thus, the value of sec(tan^-1x) is √x² + 1.