Question:

Step 1: The value of $\tan\left(\frac{x}{2}\right)$ is given as $\frac{15}{29}$. $$ \tan\left(\frac{x}{2}\right) = \frac{15}{29} $$ Step 2: Calculate $\cos x$ using the half-angle identity $\cos x = \frac{1-\tan^2(x/2)}{1+\tan^2(x/2)}$. $$ \cos x = \frac{1 - \left(\frac{15}{29}\right)^2}{1 + \left(\frac{15}{29}\right)^2} $$ $$ \cos x = \frac{1 - \frac{225}{841}}{1 + \frac{225}{841}} $$ $$ \cos x = \frac{\frac{841 - 225}{841}}{\frac{841 + 225}{841}} $$ $$ \cos x = \frac{\frac{616}{841}}{\frac{1066}{841}} $$ $$ \cos x = \frac{616}{1066} = \frac{308}{533} $$ Step 3: Calculate $\sin x$ using the identity $\sin x = \sqrt{1-\cos^2 x}$. Since $0 < x < \frac{\pi}{2}$, $\sin x$ is positive. $$ \sin x = \sqrt{1 - \left(\frac{308}{533}\right)^2} $$ $$ \sin x = \sqrt{1 - \frac{94864}{284089}} $$ $$ \sin x = \sqrt{\frac{284089 - 94864}{284089}} $$ $$ \sin x = \sqrt{\frac{189225}{284089}} $$ $$ \sin x = \frac{\sqrt{189225}}{\sqrt{284089}} $$ $$ \sin x = \frac{435}{533} $$