Final answer:
To simplify the expression sec(-t) - cos(-t)tan^2(-t), we substitute the given value of cos(t) = 2/3. Using trigonometric identities, we simplify the expression to (9 - 3(sin(t))^2)/6. Substituting sin(t) = sqrt(5)/3, we get 2/3 as the final answer.
Step-by-step explanation:
To simplify the expression sec(-t) - cos(-t)tan^2(-t), we need to use the trigonometric identities. First, we know that sec(-t) is equal to 1/cos(-t). Using the even property of cosine, cos(-t) is equal to cos(t). So sec(-t) is equal to 1/cos(t). Next, we know that cos(-t) is equal to cos(t). Finally, tan^2(-t) is equal to sin^2(-t)/cos^2(-t), which is equal to sin^2(t)/cos^2(t) since sin(-t) is equal to -sin(t) and cos(-t) is equal to cos(t). Substituting these values into the expression, we have 1/cos(t) - cos(t) * sin^2(t)/cos^2(t). This simplifies to 1/cos(t) - sin^2(t)/cos(t). Now, we can substitute the given value cos(t) = 2/3 into the expression. 1/(2/3) - (sin(t))^2/(2/3) = 3/2 - (sin(t))^2/(2/3) = (9 - 3(sin(t))^2)/6. Since cos(t) = 2/3, sin(t) = sqrt(1 - cos^2(t)) = sqrt(1 - (2/3)^2) = sqrt(1 - 4/9) = sqrt(5/9) = sqrt(5)/3. Substituting this value into the expression, we have (9 - 3(sqrt(5)/3)^2)/6 = (9 - 3(5/9))/6 = (9 - 5)/6 = 4/6 = 2/3. Therefore, the simplified and evaluated expression is 2/3.