Chapter 4.3, Problem 47E

(a)

To determine

To calculate: To confirm the identity for $x>0$

Answer to Problem 47E

The given identity is confirmed.

Explanation of Solution

Given Information: The given identity is ${\cos }^{-1}x+{\sin }^{-1}x=\frac{\pi }{2}$

Calculation:

Consider the left-hand side of the identity ${\cos }^{-1}x+{\sin }^{-1}x=\frac{\pi }{2}$ ,

  $\begin{array}{c}{\cos }^{-1}x+{\sin }^{-1}x=\left(\frac{\pi }{2}-{\sin }^{-1}x\right)+{\sin }^{-1}x\left(\because {\cos }^{-1}x=\frac{\pi }{2}-{\sin }^{-1}x\right)\\ =\frac{\pi }{2}-{\sin }^{-1}x+{\sin }^{-1}x\\ =\frac{\pi }{2}\end{array}$

The given identity is confirmed.

(b)

To determine

To calculate: To confirm the identity for $x>0$

Answer to Problem 47E

The given identity is confirmed.

Explanation of Solution

Given Information: The given identity is ${\tan }^{-1}x+{\cot }^{-1}x=\frac{\pi }{2}$

Calculation:

Consider the left-hand side of the identity ${\tan }^{-1}x+{\cot }^{-1}x=\frac{\pi }{2}$ ,

  $\begin{array}{c}{\tan }^{-1}x+{\cot }^{-1}x={\tan }^{-1}x+\left(\frac{\pi }{2}-{\tan }^{-1}x\right)\left(\because {\cot }^{-1}x=\frac{\pi }{2}-{\tan }^{-1}x\right)\\ ={\tan }^{-1}x+\frac{\pi }{2}-{\tan }^{-1}x\\ =\frac{\pi }{2}\end{array}$

The given identity is confirmed.

(c)

To determine

To calculate: To confirm the identity for $x>0$

Answer to Problem 47E

The given identity is confirmed.

Explanation of Solution

Given Information: The given identity is ${\sec }^{-1}x+{\csc }^{-1}x=\frac{\pi }{2}$

Calculation:

Consider the left-hand side of the identity ${\csc }^{-1}x+{\sec }^{-1}x=\frac{\pi }{2}$ ,

  $\begin{array}{c}{\csc }^{-1}x+{\sec }^{-1}x=\left(\frac{\pi }{2}-{\sec }^{-1}x\right)+{\sec }^{-1}x\left(\because {\csc }^{-1}x=\frac{\pi }{2}-{\sec }^{-1}x\right)\\ =\frac{\pi }{2}-{\sec }^{-1}x+{\sec }^{-1}x\\ =\frac{\pi }{2}\end{array}$

The given identity is confirmed.