Hyperbolic Functions

We need to simplify the expression

%5Ccoth%5E2%20x%20-%20%5Ctanh%5E2%20x

.
First, express

%5Ccoth%20x

and

%5Ctanh%20x

in terms of

%5Ccosh%20x

and

%5Csinh%20x

%5Ccoth%20x%20%3D%20%5Cfrac%7B%5Ccosh%20x%7D%7B%5Csinh%20x%7D

and

%5Ctanh%20x%20%3D%20%5Cfrac%7B%5Csinh%20x%7D%7B%5Ccosh%20x%7D

.
The expression becomes:

(%5Cfrac%7B%5Ccosh%20x%7D%7B%5Csinh%20x%7D)%5E2%20-%20(%5Cfrac%7B%5Csinh%20x%7D%7B%5Ccosh%20x%7D)%5E2%20%3D%20%5Cfrac%7B%5Ccosh%5E2%20x%7D%7B%5Csinh%5E2%20x%7D%20-%20%5Cfrac%7B%5Csinh%5E2%20x%7D%7B%5Ccosh%5E2%20x%7D

.
Combine the fractions by finding a common denominator:

%5Cfrac%7B%5Ccosh%5E4%20x%20-%20%5Csinh%5E4%20x%7D%7B%5Csinh%5E2%20x%20%5Ccosh%5E2%20x%7D

.
The numerator is a difference of squares:

%5Ccosh%5E4%20x%20-%20%5Csinh%5E4%20x%20%3D%20(%5Ccosh%5E2%20x%20-%20%5Csinh%5E2%20x)(%5Ccosh%5E2%20x%20%2B%20%5Csinh%5E2%20x)

.
Using the fundamental hyperbolic identities:

%5Ccosh%5E2%20x%20-%20%5Csinh%5E2%20x%20%3D%201

%5Ccosh%5E2%20x%20%2B%20%5Csinh%5E2%20x%20%3D%20%5Ccosh(2x)

.
So, the numerator simplifies to

1%20%5Ccdot%20%5Ccosh(2x)%20%3D%20%5Ccosh(2x)

.
The denominator can be related to

%5Csinh(2x)

. We know

%5Csinh(2x)%20%3D%202%20%5Csinh%20x%20%5Ccosh%20x

.
Therefore,

(%5Csinh%20x%20%5Ccosh%20x)%5E2%20%3D%20(%5Cfrac%7B%5Csinh(2x)%7D%7B2%7D)%5E2%20%3D%20%5Cfrac%7B%5Csinh%5E2(2x)%7D%7B4%7D

.
Substituting these back into the expression:

%5Cfrac%7B%5Ccosh(2x)%7D%7B%5Cfrac%7B%5Csinh%5E2(2x)%7D%7B4%7D%7D%20%3D%20%5Cfrac%7B4%5Ccosh(2x)%7D%7B%5Csinh%5E2(2x)%7D

.
This does not match any of the options directly. Let's re-evaluate the options. The provided key is (D). Let's simplify (D).

4%5Ccosh%5E2%20x%20(%5Ctext%7Bcosech%7D2x)%5E2%20%3D%204%5Ccosh%5E2%20x%20%5Cleft(%5Cfrac%7B1%7D%7B%5Csinh%202x%7D%5Cright)%5E2%20%3D%20%5Cfrac%7B4%5Ccosh%5E2%20x%7D%7B%5Csinh%5E2%202x%7D%20%3D%20%5Cfrac%7B4%5Ccosh%5E2%20x%7D%7B(2%5Csinh%20x%20%5Ccosh%20x)%5E2%7D%20%3D%20%5Cfrac%7B4%5Ccosh%5E2%20x%7D%7B4%5Csinh%5E2%20x%20%5Ccosh%5E2%20x%7D%20%3D%20%5Cfrac%7B1%7D%7B%5Csinh%5E2%20x%7D%20%3D%20%5Ccoth%5E2%20x

. This is not correct. There is an error.
Let's re-simplify the original expression in another way.

%5Ccoth%5E2%20x%20-%20%5Ctanh%5E2%20x%20%3D%20(1%2B%5Ctext%7Bcosech%7D%5E2%20x)%20-%20(1-%5Ctext%7Bsech%7D%5E2%20x)%20%3D%20%5Ctext%7Bcosech%7D%5E2%20x%20%2B%20%5Ctext%7Bsech%7D%5E2%20x

%3D%20%5Cfrac%7B1%7D%7B%5Csinh%5E2%20x%7D%20%2B%20%5Cfrac%7B1%7D%7B%5Ccosh%5E2%20x%7D%20%3D%20%5Cfrac%7B%5Ccosh%5E2%20x%20%2B%20%5Csinh%5E2%20x%7D%7B%5Csinh%5E2%20x%20%5Ccosh%5E2%20x%7D%20%3D%20%5Cfrac%7B%5Ccosh(2x)%7D%7B(%5Cfrac%7B1%7D%7B2%7D%5Csinh(2x))%5E2%7D%20%3D%20%5Cfrac%7B4%5Ccosh(2x)%7D%7B%5Csinh%5E2(2x)%7D

.
This result is consistent. All options seem to be incorrect, or there is a typo in the question or options. Let's re-examine option (D) from the image:

4%5Ccosh2x%20(%5Ctext%7Bcosech%7D2x)%5E2

. OCR was

4%5Ccosh%5E2%20x

. The image has

%5Ccosh%202x

.
Let's simplify

4%5Ccosh%202x%20(%5Ctext%7Bcosech%7D2x)%5E2

4%5Ccosh%202x%20(%5Cfrac%7B1%7D%7B%5Csinh%202x%7D)%5E2%20%3D%20%5Cfrac%7B4%5Ccosh%202x%7D%7B%5Csinh%5E2%202x%7D

.
This exactly matches my simplification. So the OCR had a typo, but the keyed answer (with the corrected term from the image) is likely correct.

Step 1: Understanding the Concept
This problem involves evaluating expressions with inverse hyperbolic functions (

%5Ctext%7BSinh%7D%5E%7B-1%7D%2C%20%5Ctext%7BCosh%7D%5E%7B-1%7D

) and then using the result in an inverse trigonometric function (

%5Ctext%7BTan%7D%5E%7B-1%7D

). We need the logarithmic definitions of the inverse hyperbolic functions.
Step 2: Key Formula or Approach
The logarithmic forms of the inverse hyperbolic functions are: 1.

%5Ctext%7BSinh%7D%5E%7B-1%7Dx%20%3D%20%5Clog(x%20%2B%20%5Csqrt%7Bx%5E2%2B1%7D)

%5Ctext%7BCosh%7D%5E%7B-1%7Dy%20%3D%20%5Clog(y%20%2B%20%5Csqrt%7By%5E2-1%7D)

for

y%20%5Cge%201

. We are given the value of these functions, so we can set up equations to solve for

x

and

y

.
Step 3: Detailed Explanation
We are given:

%5Ctext%7BSinh%7D%5E%7B-1%7Dx%20%3D%20%5Clog(1%2B%5Csqrt%7B2%7D)

%5Ctext%7BCosh%7D%5E%7B-1%7Dy%20%3D%20%5Clog(1%2B%5Csqrt%7B2%7D)

Solving for x: Using the formula for

%5Ctext%7BSinh%7D%5E%7B-1%7Dx

%5Clog(x%20%2B%20%5Csqrt%7Bx%5E2%2B1%7D)%20%3D%20%5Clog(1%2B%5Csqrt%7B2%7D)

By comparing the arguments of the logarithm:

x%20%2B%20%5Csqrt%7Bx%5E2%2B1%7D%20%3D%201%2B%5Csqrt%7B2%7D

Let's identify the form. If

x%3D1

, then

1%2B%5Csqrt%7B1%5E2%2B1%7D%20%3D%201%2B%5Csqrt%7B2%7D

. So,

x%3D1

. Solving for y: Using the formula for

%5Ctext%7BCosh%7D%5E%7B-1%7Dy

%5Clog(y%20%2B%20%5Csqrt%7By%5E2-1%7D)%20%3D%20%5Clog(1%2B%5Csqrt%7B2%7D)

By comparing the arguments of the logarithm:

y%20%2B%20%5Csqrt%7By%5E2-1%7D%20%3D%201%2B%5Csqrt%7B2%7D

We need to solve for

y

%5Csqrt%7By%5E2-1%7D%20%3D%20(1%2B%5Csqrt%7B2%7D)%20-%20y

Square both sides:

y%5E2-1%20%3D%20(1%2B%5Csqrt%7B2%7D)%5E2%20-%202y(1%2B%5Csqrt%7B2%7D)%20%2B%20y%5E2

y%5E2-1%20%3D%20(1%2B2%2B2%5Csqrt%7B2%7D)%20-%202y(1%2B%5Csqrt%7B2%7D)%20%2B%20y%5E2

-1%20%3D%203%2B2%5Csqrt%7B2%7D%20-%202y(1%2B%5Csqrt%7B2%7D)

2y(1%2B%5Csqrt%7B2%7D)%20%3D%204%2B2%5Csqrt%7B2%7D%20%3D%202(2%2B%5Csqrt%7B2%7D)

y(1%2B%5Csqrt%7B2%7D)%20%3D%202%2B%5Csqrt%7B2%7D%20%3D%20%5Csqrt%7B2%7D(%5Csqrt%7B2%7D%2B1)

y%20%3D%20%5Csqrt%7B2%7D

Calculate the final expression: We need to find

%5Ctext%7BTan%7D%5E%7B-1%7D(x%2By)

. We found

x%3D1

and

y%3D%5Csqrt%7B2%7D

. So,

x%2By%20%3D%201%2B%5Csqrt%7B2%7D

. The expression is

%5Ctext%7BTan%7D%5E%7B-1%7D(1%2B%5Csqrt%7B2%7D)

. We know that

%5Ctan(75%5E%5Ccirc)%20%3D%20%5Ctan(45%5E%5Ccirc%2B30%5E%5Ccirc)%20%3D%20%5Cfrac%7B%5Ctan%2045%5E%5Ccirc%20%2B%20%5Ctan%2030%5E%5Ccirc%7D%7B1-%5Ctan%2045%5E%5Ccirc%20%5Ctan%2030%5E%5Ccirc%7D%20%3D%20%5Cfrac%7B1%2B1%2F%5Csqrt%7B3%7D%7D%7B1-1%2F%5Csqrt%7B3%7D%7D%20%3D%20%5Cfrac%7B%5Csqrt%7B3%7D%2B1%7D%7B%5Csqrt%7B3%7D-1%7D

. Rationalizing:

%5Cfrac%7B(%5Csqrt%7B3%7D%2B1)%5E2%7D%7B(%5Csqrt%7B3%7D-1)(%5Csqrt%7B3%7D%2B1)%7D%20%3D%20%5Cfrac%7B3%2B1%2B2%5Csqrt%7B3%7D%7D%7B3-1%7D%20%3D%20%5Cfrac%7B4%2B2%5Csqrt%7B3%7D%7D%7B2%7D%20%3D%202%2B%5Csqrt%7B3%7D

. This is not

1%2B%5Csqrt%7B2%7D

. Let me check the value of

%5Ctan(67.5%5E%5Ccirc)

%5Ctan(67.5%5E%5Ccirc)%20%3D%20%5Ctan(135%5E%5Ccirc%2F2)%20%3D%20%5Cfrac%7B1-%5Ccos%20135%5E%5Ccirc%7D%7B%5Csin%20135%5E%5Ccirc%7D%20%3D%20%5Cfrac%7B1-(-1%2F%5Csqrt%7B2%7D)%7D%7B1%2F%5Csqrt%7B2%7D%7D%20%3D%20%5Cfrac%7B1%2B1%2F%5Csqrt%7B2%7D%7D%7B1%2F%5Csqrt%7B2%7D%7D%20%3D%20%5Csqrt%7B2%7D%2B1

. So,

%5Ctext%7BTan%7D%5E%7B-1%7D(1%2B%5Csqrt%7B2%7D)%20%3D%2067.5%5E%5Ccirc

. This matches one of the likely options from the OCR (

67%5Cfrac%7B1%7D%7B2%7D%5E%5Ccirc

). Step 4: Final Answer
We found

x%3D1

and

y%3D%5Csqrt%7B2%7D

. The expression to evaluate is

%5Ctext%7BTan%7D%5E%7B-1%7D(1%2B%5Csqrt%7B2%7D)

. The value of

%5Ctan(67.5%5E%5Ccirc)

1%2B%5Csqrt%7B2%7D

. Therefore,

%5Ctext%7BTan%7D%5E%7B-1%7D(1%2B%5Csqrt%7B2%7D)%20%3D%2067.5%5E%5Ccirc

High-Yield Trend

Chapter Questions
3 MCQs

Official Solution

Official Solution

Official Solution

Hyperbolic Functions

High-Yield Trend

Chapter Questions 3 MCQs

Official Solution

Official Solution

Official Solution

Chapter Questions
3 MCQs