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Prove (a) cosh2(x) − sinh2(x) = 1 and (b) 1 − tanh 2(x) = sech 2(x). solution (a) cosh2(x) − sinh2(x) = ex + e−x 2 2 − 2 = e2x + 2 + e−2x 4 − = 4 = . (b) we start with the identity proved in part (a): cosh2(x) − sinh2(x) = 1. if we divide both sides by cosh2(x), we get 1 − sinh2(x) cosh2(x) = 1 or 1 − tanh 2(x) = .
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Aroulette wheel has 38 slots in which the ball can land. two of the slots are green, 18 are red, and 18 are black. the ball is equally likely to land in any slot. the roulette wheel is going to be spun twice, and the outcomes of the two spins are independent.the probability that it lands one time on red and another time on black in any order is(a) 0.5 (b) 0.25 (c) 0.4488 (d) 0.2244
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