JEE Main 2022 · 28 Jul · Shift 2, Mathematics question 87

Let $S=\left[-\pi,\dfrac\pi2\right)-\left\{-\dfrac\pi2,-\dfrac\pi4,-\dfrac{3\pi}4,\dfrac\pi4\right\}$. Then the number of elements in the set $A=\left\{\theta\in S:\tan\theta\left(1+\sqrt5\tan(2\theta)\right)=\sqrt5-\tan(2\theta)\right\}$ is ______.