Answered: Verify the identity sec-1 x =…

Algebra & Trigonometry with Analytic Geometry

13th Edition

ISBN:9781133382119

Author:Swokowski

Publisher:Swokowski

Chapter7: Analytic Trigonometry

Section: Chapter Questions

Problem 22RE

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