4(tan(x) − cot(x)) tan2(x) − cot2(x) = 2 sin(2x) Factor the denominator and then simplify. 4(tan(x) − cot(x)) tan2(x) − cot2(x) = 4(tan(x) − cot(x)) (tan(x) + cot(x)) = tan(x) + cot(x) Use the Reciprocal Identities and simplify the compound fraction. = sin(x) cos(x) + cos(x) sin(x) = sin2(x) + cos2(x) Use a Pythagorean Identity and a Double-Angle Formula to simplify. = 1 =

Titus works at a hotel. part of his job is to keep the complimentary pitcher of water at least half full and always with ice. when he starts his shift, the water level shows 4 gallons, or 128 cups of water. as the shift progresses, he records the level of the water every 10 minutes. after 2 hours, he uses a regression calculator to compute an equation for the decrease in water. his equation is w –0.414t + 129.549, where t is the number of minutes and w is the level of water. according to the equation, after about how many minutes would the water level be less than or equal to 64 cups?