Which of the following is not a rational number. a.`sin (tan^(-1) 3 + tan^(-1).(1)/(3))` b.`cos ((pi)/(2) - sin^(-1).(3)/(4))` c.`tan ((1)/(2) cos^(-1).(sqrt5)/(3))`

Options

A. `sin (tan^(-1) 3 + tan^(-1).(1)/(3))`
B. `cos ((pi)/(2) - sin^(-1).(3)/(4))`
C. `"log"_(2) (sin ((1)/(4) sin^(-1).(sqrt63)/(8)))`
D. `tan ((1)/(2) cos^(-1).(sqrt5)/(3))`

Explanation

To determine which of the given options is not a rational number, we will analyze each expression step by step.

### Step 1: Analyze the first expression
**Expression:** \( \sin(\tan^{-1}(3) + \tan^{-1}(\frac{1}{3})) \)

Using the identity for the sine of the sum of two angles:
\[ \sin(A + B) = \sin A \cos B + \cos A \sin B \]
Let \( A = \tan^{-1}(3) \) and \( B = \tan^{-1}(\frac{1}{3}) \).

From the properties of the tangent function:
\[ \tan(A + B) = \frac{\tan A + \tan B}{1 - \tan A \tan B} = \frac{3 + \frac{1}{3}}{1 - 3 \cdot \frac{1}{3}} = \frac{3 + \frac{1}{3}}{1 - 1} = \frac{10/3}{0} \]
This implies \( A + B = \frac{\pi}{2} \).

Thus:
\[ \sin(A + B) = \sin\left(\frac{\pi}{2}\right) = 1 \]
**Conclusion:** This is a rational number.

### Step 2: Analyze the second expression
**Expression:** \( \cos\left(\frac{\pi}{2} - \sin^{-1}\left(\frac{3}{4}\right)\right) \)

Using the co-function identity:
\[ \cos\left(\frac{\pi}{2} - x\right) = \sin(x) \]
Let \( x = \sin^{-1}\left(\frac{3}{4}\right) \), then:
\[ \cos\left(\frac{\pi}{2} - \sin^{-1}\left(\frac{3}{4}\right)\right) = \sin\left(\sin^{-1}\left(\frac{3}{4}\right)\right) = \frac{3}{4} \]
**Conclusion:** This is also a rational number.

### Step 3: Analyze the third expression
**Expression:** \( \tan\left(\frac{1}{2} \cos^{-1}\left(\frac{\sqrt{5}}{3}\right)\right) \)

Let \( \theta = \cos^{-1}\left(\frac{\sqrt{5}}{3}\right) \), then:
\[ \cos \theta = \frac{\sqrt{5}}{3} \]
Using the half-angle formula for tangent:
\[ \tan\left(\frac{\theta}{2}\right) = \frac{1 - \cos \theta}{\sin \theta} \]
First, we need to find \( \sin \theta \):
\[ \sin^2 \theta + \cos^2 \theta = 1 \implies \sin^2 \theta = 1 - \left(\frac{\sqrt{5}}{3}\right)^2 = 1 - \frac{5}{9} = \frac{4}{9} \implies \sin \theta = \frac{2}{3} \]
Now substituting back:
\[ \tan\left(\frac{\theta}{2}\right) = \frac{1 - \frac{\sqrt{5}}{3}}{\frac{2}{3}} = \frac{3 - \sqrt{5}}{2} \]
**Conclusion:** This expression \( \frac{3 - \sqrt{5}}{2} \) is not a rational number because \( \sqrt{5} \) is irrational.

### Final Answer
The option that is not a rational number is:
**c.** \( \tan\left(\frac{1}{2} \cos^{-1}\left(\frac{\sqrt{5}}{3}\right)\right) \)

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