Answered: Verify the identity sec-1 x =…

Calculus: Early Transcendentals

8th Edition

ISBN:9781285741550

Author:James Stewart

Publisher:James Stewart

Chapter1: Functions And Models

Section: Chapter Questions

Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...

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Question

Verify the identity sec^-1 x = cos^-1(1/x), for x ≠ 0.

Expert Solution

Step 1

According to the question, we have to verify the identity $\sec^{- 1} (x) = \cos^{- 1} (\frac{1}{x})$ .

As the above function is about inverse trigonometric function and the inverse trigonometric functions are the inverse functions of the trigonometric functions.

We can verify the above function using suitable identity and solve accordingly.

Step 2

Rewrite the given expression,

$\sec^{- 1} (x) = \cos^{- 1} (\frac{1}{x})$

Now, taking left hand side and substitute $\sec^{- 1} x = t$ and solve further,

$\sec^{- 1} x = t x = sect$

As we know that sec is inverse of cos,so we can solve as follows,

$x = \frac{1}{\cos t} \cos t = \frac{1}{x}$

And also the above function can be written as follows,

$t = \cos^{- 1} (\frac{1}{x}) \dots (1)$

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