4950 The function cot1 x is defined to be the inverse ofthe restricted cotangent

Chapter 0, Problem 50

(choose chapter or problem)

The function \(\cot ^{-1} x\) is defined to be the inverse of the restricted cotangent function \(\cot x, 0<x<\pi\) and the function \(\csc ^{-1} x\) is defined to be the inverse of the restricted cosecant function \(\csc x,-\pi / 2<x<\pi / 2, x \neq 0\).Use these definitions in these and in all subsequent exercises that involve these functions.

Show that

(a) \(\cot ^{-1} x=\left\{\tan ^{-1}(1 / x)\right\}\), if \(x>0 \pi+\tan ^{-1}(1 / x)\), if \(x<0\)

(b) \(\sec ^{-1} x=\cos ^{-1} \frac{1}{x}\), if \(|x| \geq 1\)

(c) \(\csc ^{-1} x=\sin ^{-1} \frac{1}{x}\), if \(|x| \geq 1\)

Equation Transcription:

Text Transcription:

cot^-1 ⁡x

cot x,0<x<pi

csc^-1 x

csc x,-pi/2<x<pi/2, x not =0

cot^-1x={tan^-1(1/x)}

x>0 pi+tan-1(1/x)

x<0

sec^-1 x=cos^-1 1/x

|x|1

csc^-1x=sin^-1 1/x