Question:

Step 1: Evaluate $\sec^2(\tan^{-1}2)$ using the identity relating secant and tangent. Let $\alpha = \tan^{-1}2$, which means $\tan\alpha = 2$. Using the identity $\sec^2\alpha = 1 + \tan^2\alpha$: $$\sec^2\alpha = 1 + 2^2 = 1 + 4 = 5$$ Step 2: Evaluate $\csc^2(\cot^{-1}3)$ using the identity relating cosecant and cotangent. Let $\beta = \cot^{-1}3$, which means $\cot\beta = 3$. Using the identity $\csc^2\beta = 1 + \cot^2\beta$: $$\csc^2\beta = 1 + 3^2 = 1 + 9 = 10$$ Step 3: Find the sum of the first two parts. $$5 + 10 = 15$$ Step 4: Evaluate $\csc(2\cot^{-1}2 + \cos^{-1}\frac{3}{5})$ by first finding the trigonometric values for $\cot^{-1}2$. Let $\phi = \cot^{-1}2$, so $\cot\phi = 2$, which means $\tan\phi = \frac{1}{2}$. From a right triangle with opposite side 1 and adjacent side 2, the hypotenuse is $\sqrt{1^2 + 2^2} = \sqrt{5}$. Therefore: $$\sin\phi = \frac{1}{\sqrt{5}}, \quad \cos\phi = \frac{2}{\sqrt{5}}$$ Step 5: Calculate $\sin 2\phi$ and $\cos 2\phi$ using double angle formulas. $$\sin 2\phi = 2\sin\phi\cos\phi = 2 \cdot \frac{1}{\sqrt{5}} \cdot \frac{2}{\sqrt{5}} = \frac{4}{5}$$ $$\cos 2\phi = \cos^2\phi - \sin^2\phi = \frac{4}{5} - \frac{1}{5} = \frac{3}{5}$$ Step 6: Find the trigonometric values for $\gamma = \cos^{-1}\frac{3}{5}$. Let $\gamma = \cos^{-1}\frac{3}{5}$, so $\cos\gamma = \frac{3}{5}$. From a right triangle with adjacent side 3 and hypotenuse 5, the opposite side is $\sqrt{5^2 - 3^2} = 4$. Therefore: $$\sin\gamma = \frac{4}{5}$$ Step 7: Calculate $\sin(2\phi + \gamma)$ using the angle addition formula. $$\sin(2\phi + \gamma) = \sin 2\phi \cos\gamma + \cos 2\phi \sin\gamma$$ $$= \frac{4}{5} \cdot \frac{3}{5} + \frac{3}{5} \cdot \frac{4}{5}$$ $$= \frac{12}{25} + \frac{12}{25} = \frac{24}{25}$$ Step 8: Find $\csc(2\phi + \gamma)$ and calculate the final answer. $$\csc(2\phi + \gamma) = \frac{1}{\sin(2\phi + \gamma)} = \frac{1}{\frac{24}{25}} = \frac{25}{24}$$ The total value of the expression is: $$15 + \frac{25}{24} = 24 + \frac{15}{25}$$ **Final Answer: The value of the expression is $24 + \frac{15}{25}$, which corresponds to Option 2.**