If θ is an acute angle and tan θ + cot θ = 2, then find the value of tan3 θ + cot3 θ + 6 tan3 θ cot2 θ.
Show answer & explanation
Correct answer: D
Explanation
The correct answer is option 4 i.e. 8.
tanθ + cotθ = 2 ......(given)
so
tanθ = x and cotθ = 1/x
⇒ x + 1/x = 2
⇒ x2 + 1 = 2x
⇒ x2 − 2x + 1 = 0
⇒ (x−1)2 = 0
Thus, x = 1
tan θ = 1 and cot θ = 1
so tan3θ + cot3θ + 6tan3θ cot2θ = 13 + 13 + 6 × 13 × 12 = 1 + 1 + 6 = 8