Which of the following are identities check all that apply a. cos(x+y)-cos(x-y)=2cosxcosy b. sin(x+y)+sin(x-y)=2cosxsiny c. cos(x+y)+cos(x-y)=2cosxcosy d. tan(x-pi)=tanx

Arandom sample of n = 25 observations is taken from a n(µ, σ ) population. a 95% confidence interval for µ was calculated to be (42.16, 57.84). the researcher feels that this interval is too wide. you want to reduce the interval to a width at most 12 units. a) for a confidence level of 95%, calculate the smallest sample size needed. b) for a sample size fixed at n = 25, calculate the largest confidence level 100(1 − α)% needed.