Respuesta :
Answer:
Read explanation
Step-by-step explanation:
Given cot(u) = 2, we can use trigonometric identities to find the exact values of the given expressions:
A) To find the exact value of tan(u), we can use the reciprocal identity:
tan(u) = 1 / cot(u)
Substituting the given value cot(u) = 2, we have:
tan(u) = 1 / 2
Therefore, the exact value of tan(u) is 1/2.
B) To find the exact value of csc^2(u), we can use the Pythagorean identity:
csc^2(u) = 1 + cot^2(u)
Substituting the given value cot(u) = 2, we have:
csc^2(u) = 1 + (2)^2
= 1 + 4
= 5
Therefore, the exact value of csc^2(u) is 5.
C) To find the exact value of tan(u/2 - u), we can use the half-angle identity:
tan(u/2 - u) = -tan(3u/2)
Substituting the given value cot(u) = 2, we can rewrite tan(3u/2) in terms of cot(u):
tan(3u/2) = cot(u) / tan(u)
Substituting the given value cot(u) = 2, we have:
tan(u/2 - u) = -2 / tan(u)
Since we already found that tan(u) = 1/2, we can substitute this value:
tan(u/2 - u) = -2 / (1/2)
= -4
Therefore, the exact value of tan(u/2 - u) is -4.
D) To find the exact value of sec^2(u), we can use the Pythagorean identity:
sec^2(u) = 1 + tan^2(u)
Substituting the given value tan(u) = 1/2, we have:
sec^2(u) = 1 + (1/2)^2
= 1 + 1/4
= 5/4
Therefore, the exact value of sec^2(u) is 5/4.
The correct answers are:
A) tan(u) = 1/2
B) csc^2(u) = 5
C) tan(u/2 - u) = -4
D) sec^2(u) = 5/4.
