Simplifying the Expression cosh^2(x) – sinh^2(x) | The Definition and Calculation of Hyperbolic Trigonometric Functions

cosh^2(x) – sinh^2(x)

To simplify the expression cosh^2(x) – sinh^2(x), we first need to understand the definitions of cosh(x) and sinh(x)

To simplify the expression cosh^2(x) – sinh^2(x), we first need to understand the definitions of cosh(x) and sinh(x).

Cosh(x) stands for hyperbolic cosine, and sinh(x) represents hyperbolic sine. These functions are part of hyperbolic trigonometry, which is an extension of regular trigonometry that deals with the hyperbolic functions based on the unit hyperbola.

The hyperbolic functions are defined as follows:

cosh(x) = (e^x + e^(-x))/2
sinh(x) = (e^x – e^(-x))/2

Now, let’s substitute these definitions into the original expression:

cosh^2(x) – sinh^2(x) = [(e^x + e^(-x))/2]^2 – [(e^x – e^(-x))/2]^2

To simplify this, we can use the identity (a + b)^2 = a^2 + 2ab + b^2.

Applying this identity, we get:

[(e^x + e^(-x))/2]^2 – [(e^x – e^(-x))/2]^2
= [(e^x)^2 + 2(e^x)(e^(-x)) + (e^(-x))^2]/4 – [(e^x)^2 – 2(e^x)(e^(-x)) + (e^(-x))^2]/4

Notice that the terms (e^x)(e^(-x)) and -2(e^x)(e^(-x)) have opposite signs, so they will cancel out when subtracted.

Simplifying further, we have:

= [(e^x)^2 + 2(e^x)(e^(-x)) + (e^(-x))^2 – (e^x)^2 + 2(e^x)(e^(-x)) – (e^(-x))^2]/4
= [4(e^x)(e^(-x))]/4
= e^x * e^(-x)

Using the rule of exponents e^a * e^b = e^(a+b), we can simplify e^x * e^(-x) to e^(x+(-x)) = e^0 = 1.

Therefore, cosh^2(x) – sinh^2(x) simplifies to:

cosh^2(x) – sinh^2(x) = 1

So, no matter what value of x we choose, the expression evaluates to 1.

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