If tan θ + cot θ = 2 and θ is an acute angle, find the value of tan³θ + cot³θ.
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Correct answer: C
Explanation
Given:
tanθ + cotθ = 2
θ is an acute angle
By the formula:
(a + b)³ = a³ + b³ + 3ab(a + b)
tanθ × cotθ = 1
Let
a = tanθ, b = cotθ
a + b = 2
ab = 1
(a + b)³ = a³ + b³ + 3ab(a + b)
2³ = a³ + b³ + 3 × 1 × 2
8 = a³ + b³ + 6
a³ + b³ = 2