`a^3/2 cosec^2(1/2 tan^(-1)""a/b)+b^3/2 sec^2(1/2tan^(-1)""b/a)` is equal to

Options

A. `(a-b)(a^2+b^2)`
B. `(a+b)(a^2-b^2)`
C. `(a+b)(a^2+b^2)`
D. none of these

Answer

To solve the expression \( \frac{a^{3/2}}{\csc^2\left(\frac{1}{2} \tan^{-1}\left(\frac{a}{b}\right)\right)} + \frac{b^{3/2}}{\sec^2\left(\frac{1}{2} \tan^{-1}\left(\frac{b}{a}\right)\right)} \), we will follow these steps:

### Step 1: Convert Cosecant and Secant
We know that:
\[ \csc^2 \theta = \frac{1}{\sin^2 \theta} \quad \text{and} \quad \sec^2 \theta = \frac{1}{\cos^2 \theta} \]
Thus, we can rewrite the expression as:
\[ \frac{a^{3/2} \sin^2\left(\frac{1}{2} \tan^{-1}\left(\frac{a}{b}\right)\right)}{1} + \frac{b^{3/2} \cos^2\left(\frac{1}{2} \tan^{-1}\left(\frac{b}{a}\right)\right)}{1} \]

### Step 2: Use the Half-Angle Formulas
Using the half-angle formulas:
\[ \sin^2 \theta = \frac{1 - \cos(2\theta)}{2} \quad \text{and} \quad \cos^2 \theta = \frac{1 + \cos(2\theta)}{2} \]
We can express:
\[ \sin^2\left(\frac{1}{2} \tan^{-1}\left(\frac{a}{b}\right)\right) = \frac{1 - \cos\left(\tan^{-1}\left(\frac{a}{b}\right)\right)}{2} \]
\[ \cos^2\left(\frac{1}{2} \tan^{-1}\left(\frac{b}{a}\right)\right) = \frac{1 + \cos\left(\tan^{-1}\left(\frac{b}{a}\right)\right)}{2} \]

### Step 3: Find the Cosine Values
Using the definition of tangent:
\[ \cos\left(\tan^{-1}\left(\frac{a}{b}\right)\right) = \frac{b}{\sqrt{a^2 + b^2}} \quad \text{and} \quad \cos\left(\tan^{-1}\left(\frac{b}{a}\right)\right) = \frac{a}{\sqrt{a^2 + b^2}} \]

### Step 4: Substitute Back
Substituting these values back into the expression:
\[ \frac{a^{3/2} \left(\frac{1 - \frac{b}{\sqrt{a^2 + b^2}}}{2}\right)}{1} + \frac{b^{3/2} \left(\frac{1 + \frac{a}{\sqrt{a^2 + b^2}}}{2}\right)}{1} \]

### Step 5: Simplify the Expression
Now, simplifying:
\[ = \frac{a^{3/2}}{2} \left(1 - \frac{b}{\sqrt{a^2 + b^2}}\right) + \frac{b^{3/2}}{2} \left(1 + \frac{a}{\sqrt{a^2 + b^2}}\right) \]
Combine the terms:
\[ = \frac{1}{2} \left(a^{3/2} + b^{3/2} - \frac{a^{3/2} b}{\sqrt{a^2 + b^2}} + \frac{a b^{3/2}}{\sqrt{a^2 + b^2}}\right) \]

### Step 6: Factor Out Common Terms
Factoring out common terms:
\[ = \frac{1}{2} \left(a^{3/2} + b^{3/2} + \frac{ab(a^{1/2} - b^{1/2})}{\sqrt{a^2 + b^2}}\right) \]

### Final Result
Thus, the final simplified expression is:
\[ \frac{a^{3/2} + b^{3/2} + ab(a^{1/2} - b^{1/2})}{2\sqrt{a^2 + b^2}} \]