At age 30 you deposit $150 at the end of each month into an IRA that pays 4% interest compounded monthly. In the previous question, we found that the value of the annuity at age 65 would be $137,059.64. How much interest did you earn? Round your answer to the nearest cent and do not include the dollar sign.

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) = .