Use Osborne's rule to write down the hyperbolic counterparts of the following trigonometric identities. 1. sinx = (1 - cos 2x) 3. cosx = (1 + cos2x) 5. tan 2x = 13. 1 2 tan x Use the definitions of the hyperbolic functions to prove that: 7. 2 sinh x cosh x = sinh 2x 9, tanh(lnv3) = = 11. (coth x) = -cosechx dx 1 + tanh ax 1 - tanh 4x 15. tanx 1 = ex 1 tanh2 x 1 + tanh2x 17. cosh*t - sinh't = cosh 2t 2. sin(x - y) = sin x cos y cos x sin y 4. cos (x - y) = cos x cos y + sin x sin y 6. sin 3t = 3 sint - 4 sint = sech 2x 8. cosh x cosh y + sinh x sinh y = cosh(x + y) 10. 4 cosht = cosh 3t+ 3 cosht 12. (sech x) = -sech x tanh x Make use of the basic hyperbolic identities (the hyperbolic counterparts of the trigonometric identities listed in section 1.1) to prove the following identities. 14. sinh 2z 1 + cosh 2z = tanh z 16. cosh x + tanh x sinhxtanhx-1 sech x + tanh x 18. coth 2u 1 - cosech 2u = tanh u