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Hyperbolic Functions

In mathematics, hyperbolic functions are analogs of the ordinary trigonometric, or circular functions.

The basic hyperbolic functions are the hyperbolic sine "sinh" and the hyperbolic cosine "cosh" from which are derived the hyperbolic tangent "tanh" hyperbolic cosecant "csch" or "cosech" or hyperbolic secant "sech" and hyperbolic cotangent "coth" corresponding to the derived trigonometric functions.

Some formulas are as given below

1. $$Cosh^2 θ - Sinn^2 θ = 1$$
2. $$Sech^2 θ + tanh^2 θ = 1$$
3. $$Coth^2 θ - Cosech^2 θ = 1$$
4. $$Sinh (A + B) = Sinh A.Cosh B + Cosh A.Sinh B$$
5. $$Sinh (A - B) = Sinh A. Cosh B - Cosh A. Sinh B $$
6. $$Cosh (A - B) = Cosh A Cosh B + Sinh A Sinh B$$
7. $$Cosh (A - B) = Cosh A Cosh B - Sinh A Sinh B$$
8. $$tanh (A + B) = {tanh A - tanh B \over 1 - tanh A tanh B}.$$
9. $$tanh (A - B) = {tanh A - tanh B \over 1 + tanh A tanh B}.$$
10. $$sinh 2A = 2 sinh A . Cosh B $$
    & $$ sinh 2A = {2tanh A \over 1 - tanh^2 A}.$$
    
11. $$cosh 2A = cosh^2 A + sinh^2 A$$
    $$cosh 2A = 1 + 2 sinh^2 A$$
    $$cosh 2A = 2cosh^2 A - 1$$
    $$cosh 2A = 2cosh^2 A - 1$$ $$cosh 2A = {1 + tanh^2 A \over 1 - tanh^2 A}.$$
    OR
    $$cosh 2A + 1 = 2 cosh^2 A$$
    OR
    $$cosh 2A - 1 = 2sinh^2 A$$
12. $$tanh 2A = {2 tanh A\over 1 + tanh^2 A}.$$
13. $$sinh 3A = 3 sinh A + 4 sinh^3 A$$
14. $$cosh 3A = 4cosh^3 A - 3cosh A$$
15. $$tanh 3A = {3tanh A + tanh^3 B\over 1 + 3tanh^2 A}.$$
16. $$sinh (A + B) + sinh (A - B) = 2sinh A cosh B$$
17. $$sinh (A + B) - sinh (A - B) = 2cosh A sinh B$$
18. $$cosh (A + B) + cosh (A - B) = 2cosh A cosh B$$
19. $$cosh (A + B) - cosh (A - B) = 2sinh A sinh B$$
20. $$sinh A + sinh B = 2sinh ({A + B \over 2}) cosh ({A - B \over 2})$$
21. $$sinh A - sinh B = 2cosh ({A + B \over 2}) sinh ({A - B \over 2})$$
22. $$cosh A + cosh B = 2cosh ({A + B \over 2}) cosh ({A - B \over 2})$$
23. $$cosh A - cosh B = 2sinh ({A + B \over 2}) sinh B ({A - B \over 2})$$

Mr.kale Gorkhashnath
B.A. D.ed
Sandesh Vidyalaya, Suryanagar, Vikhroli(w), Mumbai-400083