Learn How Do You Simplify Sectan 1x

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

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

## Answer: The value of sec(tan^{-1}x) is √x^{2} + 1.

Let's solve the following.

**Explanation:**

To Simplify: sec(tan^{-1}x)

Let,

y = tan^{-1}x

⇒ x = tan y

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

Squaring on both the sides,

⇒ x^{2} = sin^{2}y / cos^{2}y

Adding 1 on both the sides of the equation,

⇒ x^{2} + 1 = (sin^{2}y / cos^{2}y) + 1

⇒ x^{2} + 1 = (sin^{2}y + cos^{2}y) / cos^{2}y

⇒ x^{2} + 1 = 1 / cos^{2}y (Since, sin^{2}y + cos^{2}y = 1)

⇒ x^{2} + 1 = sec^{2}y

Taking square root on both the sides,

⇒ √x^{2} + 1 = sec^{2}y

⇒ √x^{2} + 1 = sec(tan^{-1}x) (Since, y = tan^{-1}x)

### Thus, the value of sec(tan^{-1}x) is √x^{2} + 1.

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