For the spinner above, the probability of landing on black is 1/2 and the probability of landing on red is 1/3 Suppose it is spun three times. What is the probability of the outcome (black, white, red)?

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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) = .

Find the area of the triangle formed by the origin and the points of intersection of parabolas y=−3x^2+20 and y=x^2−16.