Hyperbolic Functions Definitions Explained

To match the hyperbolic functions in Group - 1 with their definitions in Group - 2, we first recall the fundamental definitions based on exponential functions:

• $ \sinh x = \frac{e^x - e^{-x}}{2} $
• $ \cosh x = \frac{e^x + e^{-x}}{2} $

Deriving Hyperbolic Function Definitions

Using the basic definitions, we can derive the forms for the other functions:

Matching P: $ \tanh x $

The definition is $ \tanh x = \frac{\sinh x}{\cosh x} $. Substituting the exponential forms:

$ \tanh x = \frac{\frac{e^x - e^{-x}}{2}}{\frac{e^x + e^{-x}}{2}} = \frac{e^x - e^{-x}}{e^x + e^{-x}} $

This matches Group - 2 definition **IV**. So, P $\rightarrow$ IV.

Matching Q: $ \coth x $

The definition is $ \coth x = \frac{\cosh x}{\sinh x} $. Substituting the exponential forms:

$ \coth x = \frac{\frac{e^x + e^{-x}}{2}}{\frac{e^x - e^{-x}}{2}} = \frac{e^x + e^{-x}}{e^x - e^{-x}} $

This matches Group - 2 definition **I**. So, Q $\rightarrow$ I.

Matching R: $ \text{sech } x $

The definition is $ \text{sech } x = \frac{1}{\cosh x} $. Substituting the exponential form:

$ \text{sech } x = \frac{1}{\frac{e^x + e^{-x}}{2}} = \frac{2}{e^x + e^{-x}} $

This matches Group - 2 definition **II**. So, R $\rightarrow$ II.

Matching S: $ \text{cosech } x $

The definition is $ \text{cosech } x = \frac{1}{\sinh x} $. Substituting the exponential form:

$ \text{cosech } x = \frac{1}{\frac{e^x - e^{-x}}{2}} = \frac{2}{e^x - e^{-x}} $

This matches Group - 2 definition **III**. So, S $\rightarrow$ III.

Final Combination

Combining the matches:

• P matches with IV
• Q matches with I
• R matches with II
• S matches with III

The correct combination is P-IV, Q-I, R – II, S – III.