If θ is an acute angle and cot θ + tan θ = 2, then find the value of tan12 θ + cot12 θ + 2 tan5 θ cot7 θ.
Show answer & explanation
Correct answer: A
Explanation
The correct answer is Option 1 i.e. 4.
Let tanθ = x
cotθ + tanθ = 2 [tanθ = 1/cotθ]
⇒ 1/x + x = 2
⇒ x2 + 1 = 2x
⇒ x2 - 2x + 1 = 0
⇒ x2 - 1x - 1x + 1 = 0
⇒ (x - 1) - 1(x - 1) = 0
⇒ (x - 1)(x - 1) = 0
⇒ (x - 1)2 = 0
⇒ x - 1 = 0
⇒ x = 1
So,
tanθ = 1 and, cotθ = 1/1 = 1
Now, the value of tan12 θ + cot12 θ + 2 tan5 θ cot7 θ
⇒ (1)12 + (1)12 + 2(1)5(1)7
⇒ 1 + 1 + 2 = 4