The value of \( \tan\left(\displaystyle\frac{1}{2}\cos^{-1}\displaystyle\frac{\sqrt{5}}{3}\right) \) is equal to
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → \( \displaystyle\frac{3-\sqrt{5}}{2} \)
Let \( \theta=\cos^{-1}\frac{\sqrt{5}}{3} \), so \( \cos\theta=\frac{\sqrt{5}}{3} \) and \( 0\lt\frac{\theta}{2}\lt\frac{\pi}{4} \), which makes \( \tan\frac{\theta}{2} \) positive. By the half-angle formula,
\( \displaystyle \tan\frac{\theta}{2}=\sqrt{\frac{1-\cos\theta}{1+\cos\theta}} \)
\( \displaystyle =\sqrt{\frac{3-\sqrt{5}}{3+\sqrt{5}}} \)
Multiplying inside the root by \( \frac{3-\sqrt{5}}{3-\sqrt{5}} \):
\( \displaystyle \tan\frac{\theta}{2}=\sqrt{\frac{(3-\sqrt{5})^2}{9-5}} \)
\( \displaystyle =\frac{3-\sqrt{5}}{2} \)