Evaluate: tan2[12sin−1(23)]
Let sin−1(23)=θ where θ∈[−π2,π2]
Then, sin(θ)=23
∴
Now, \; \sin \left(\theta\right) = \dfrac{2 \tan \left(\dfrac{\theta}{2}\right)}{1 + \tan^2 \left(\dfrac{\theta}{2}\right)}
i.e. \; \dfrac{2 \tan \left(\dfrac{\theta}{2}\right)}{1 + \tan^2 \left(\dfrac{\theta}{2}\right)} = \dfrac{2}{3}
i.e. \; \tan^2 \left(\dfrac{\theta}{2}\right) - 3 \tan \left(\dfrac{\theta}{2}\right) + 1 = 0
i.e. \; \tan \left(\dfrac{\theta}{2}\right) = \dfrac{3 \pm \sqrt{9 - 4}}{2}
i.e. \; \tan \left(\dfrac{\theta}{2}\right) = \dfrac{3 + \sqrt{5}}{2} \; OR \; \tan \left(\dfrac{\theta}{2}\right) = \dfrac{3 - \sqrt{5}}{2}
\therefore \; \tan^2 \left(\dfrac{\theta}{2}\right) = \left(\dfrac{3 + \sqrt{5}}{2}\right)^2 \; OR \; \tan^2 \left(\dfrac{\theta}{2}\right) = \left(\dfrac{3 - \sqrt{5}}{2}\right)^2
i.e. \; \tan^2 \left(\dfrac{\theta}{2}\right) = \dfrac{14 + 6 \sqrt{5}}{4} \; OR \; \tan^2 \left(\dfrac{\theta}{2}\right) = \dfrac{14 - 6 \sqrt{5}}{4}
i.e. \; \tan^2 \left(\dfrac{\theta}{2}\right) = \dfrac{7 \pm 3 \sqrt{5}}{2}
\because \; \dfrac{7 + 3 \sqrt{5}}{2} \; \notin \; \left[-1, 1\right] \; and \; \dfrac{7 - 3 \sqrt{5}}{2} \; \in \; \left[-1, 1\right],
\therefore \; \tan^2 \left(\dfrac{\theta}{2}\right) = \dfrac{7 - 3 \sqrt{5}}{2}
i.e. \; \tan^2 \left[\dfrac{1}{2} \sin^{-1} \left(\dfrac{2}{3}\right)\right] = \dfrac{7 - 3 \sqrt{5}}{2}