If $\sqrt{3}tan^2 θ - 4 tan θ +\sqrt{3} = 0 $, then what is the value of $tan^2 θ +cot^2 θ $ ?
Answer & explanation
Correct answer: option 2
We are given that :-
√3 tan²θ - 4 tanθ + √3 = 0
√3 tan²θ - 3 tanθ - tanθ + √3 = 0
√3 tanθ ( tanθ - √3 ) - 1 ( tanθ - √3 ) = 0
(√3 tanθ - 1 ) .( tanθ - √3 ) = 0
Either (√3 tanθ - 1 ) = 0 or ( tanθ - √3 ) = 0
So, θ = 30º or 60º
Now,
tan²θ + cot²θ
Let us assume that θ = 30º
= tan²30º + cot²30º
= \(\frac{1}{3}\) + 3
= \(\frac{10}{3}\)