How to Solve Hyperbolic Functions?

There are 6 hyperbolic Functions namely: sine h y, cosine h y, tangent h y, cosecant h y, secant h y and cotangent h y. These functions are expressed in terms of exponential functions. Let us consider some examples to learn how to solve Hyperbolic Functions. Before that we need to be aware of few formulas that are given below:

Sin h y = (e^y - e^-y) / 2,

Cos h y = (e^y + e^-y) / 2,

Tan h y = (e^y - e^-y) / (e^y + e^-y),

Cosec h y = 1 / Sin h y,

Or Cosec h y = 1 / ((e^y - e^-y) / 2) = 2 / (e^y - e^-y),

Sec h y = 1 / Cos h y,

Or Sec h y = 1 / ((e^y + e^-y) / 2) = 2 / (e^y + e^-y),

Cot h y = 1 / Tan h y = Cos h y /Sin h y,

Or Cot h y = 1 / ((e^y - e^-y) / (e^y + e^-y)) = (e^y + e^-y) / (e^y - e^-y),

Example: Solve sin2 4 h + cos 2 4 h?

Solution: Using the values for hyperbolic functions sine and cosine and y = 4, we can write:
sin² 4 h + cos ² 4 h = (e⁴ - e^-4)² /4 + (e⁴ + e^-4)² /4 = e⁸ + e^{-8 -2} + e⁸ + e^{-8 + 2} /4 = e⁸ + e^-8 /2.

Example: Solve (1 + tan² 10 h)?

Solution: Using the value of hyperbolic function tan from above, we can write:
1 + tan² 10 h = 1 + ((e¹⁰ - e^-10) / (e¹⁰ + e^-10)) = ((e¹⁰ + e^-10) + (e¹⁰ - e^-10)) / (e¹⁰ + e^-10),
= 2e10 / (e¹⁰ + e^-10).