Prove: (a) cosh ^-1 x=ln (x+√(x^2-1)), x ≥ 1 (b) tanh ^-1 x=(1)/(2)ln((1+x)/(1-x)),

step 1 problem 6y is equal to cosine inverse x x is equal to cosine y which is equal to e power y plus

Prove: (a) cosh ^-1 x=ln (x+√(x^2-1)), x ≥ 1 ...

Prove:$$
\begin{array}{l}{\text { (a) } \cosh ^{-1} x=\ln (x+\sqrt{x^{2}-1}), \quad x \geq 1} \\ {\text { (b) } \tanh ^{-1} x=\frac{1}{2} \ln \left(\frac{1+x}{1-x}\right), \quad-1<x<1}\end{array}
$$

Key Concepts

Inverse Hyperbolic Functions

Inverse hyperbolic functions are defined as the inverses of the hyperbolic functions. They allow one to express the hyperbolic angle corresponding to a given hyperbolic function value. These functions can be explicitly written in terms of logarithms, thereby establishing a connection between hyperbolic and logarithmic expressions that is fundamental in many areas of mathematics, including calculus and hyperbolic geometry.

Exponential Representation of Hyperbolic Functions

Hyperbolic functions, such as cosh and tanh, can be expressed in terms of exponential functions. For instance, the hyperbolic cosine is given by (e^y + e^(-y))/2 and the hyperbolic tangent by (e^y - e^(-y))/(e^y + e^(-y)). This exponential formulation is critical for deriving their inverses and other properties, as it translates the problem into one of solving equations involving exponentials.

Logarithmic Identities and Their Use in Inverses

Logarithmic identities play a key role in transforming the equations arising from the exponential definitions of hyperbolic functions. When solving for the inverse hyperbolic functions, logarithms are used to unravel the exponential relationships, thereby providing closed-form expressions. These identities and manipulations demonstrate the deep interplay between exponential and logarithmic functions in analysis.

Algebraic Manipulation and Solving Equations

The process of proving identities for inverse hyperbolic functions involves algebraic manipulation, including solving quadratic equations and isolating terms. By translating the hyperbolic equations into algebraic ones using substitutions and the quadratic formula, one can systematically solve for the desired expression. This technique is broadly applicable across various areas of algebra and calculus.

Transcript

Please give Ace some feedback