Question

Let $\begin{array}{l} S=\left[-\pi, \frac{\pi}{2}\right)-\left[-\frac{\pi}{2},-\frac{\pi}{4},-\frac{3\pi}{4},\frac{\pi}{4}\right].\end{array}$Then the number of elements in the set \begin{array}{l} A=\left\\{\theta\in S:\tan\theta\left(1+\sqrt{5}\tan\left(2\theta\right)\right)=\sqrt{5}-\tan\left(2\theta\right) \right\\} \end{array}is ________.

Answer 1

Here, $\begin{array}{l} tan~\alpha =\sqrt{5}\end{array}$

$\begin{array}{l} \therefore\ \tan\theta=\frac{\tan\alpha-\tan2\theta}{1+\tan\alpha\tan2\theta} \end{array}$

∴ tan θ = tan (α – 2θ)

$α – 2θ = nπ + θ$

⇒ $3θ = α – nπ$

$\begin{array}{l} \Rightarrow\ \theta = \frac{\alpha}{3}-\frac{n\pi}{3}~;~n\in Z\end{array}$

If θ [–π, π/2) then

n = 0, 1, 2, 3, 4 are acceptable

∴ 5 solutions.