What is the value of `(tan 9^(@) tan 23^(@) tan 60^(@) tan67^(@) tan 81^(@))/("cosec"^(2)72^(@)+cos^(2)15^(@)-tan^(2)18^(@)+cos^(2)75^(@))?`
(a)`(1)/(2 sqrt(3))`
(b)`(sqrt(3))/(2)`
(c)`(1)/(sqrt(3))`
(d)`2 sqrt(3)`

Options

A. `(1)/(2 sqrt(3))`
B. `(sqrt(3))/(2)`
C. `(1)/(sqrt(3))`
D. `2 sqrt(3)`

Answer

To solve the given problem, we need to evaluate the expression:

\[ \frac{\tan 9^\circ \tan 23^\circ \tan 60^\circ \tan 67^\circ \tan 81^\circ}{\csc^2 72^\circ + \cos^2 15^\circ - \tan^2 18^\circ + \cos^2 75^\circ} \]

### Step 1: Simplifying the numerator
We start with the numerator:

\[ \tan 9^\circ \tan 23^\circ \tan 60^\circ \tan 67^\circ \tan 81^\circ \]

Using the identity \(\tan(90^\circ - x) = \cot x\), we can rewrite:

- \(\tan 67^\circ = \cot 23^\circ\)
- \(\tan 81^\circ = \cot 9^\circ\)

Thus, we can pair the terms:

\[ \tan 9^\circ \tan 81^\circ = 1 \quad \text{and} \quad \tan 23^\circ \tan 67^\circ = 1 \]

This simplifies the numerator to:

\[ \tan 60^\circ = \sqrt{3} \]

### Step 2: Simplifying the denominator
Now, we simplify the denominator:

\[ \csc^2 72^\circ + \cos^2 15^\circ - \tan^2 18^\circ + \cos^2 75^\circ \]

Using the identity \(\csc^2 x = 1 + \cot^2 x\):

\[ \csc^2 72^\circ = 1 + \cot^2 72^\circ \]

Also, we know that:

- \(\cos^2 75^\circ = \sin^2 15^\circ\)
- \(\tan^2 18^\circ = \frac{\sin^2 18^\circ}{\cos^2 18^\circ}\)

Using the identity \(\sin^2 x + \cos^2 x = 1\):

\[ \cos^2 15^\circ + \sin^2 75^\circ = 1 \]

Thus, the denominator simplifies to:

\[ 1 + \cot^2 72^\circ - \tan^2 18^\circ + 1 \]

Using the identity \(\sec^2 x - \tan^2 x = 1\):

\[ \cot^2 72^\circ = \frac{1}{\tan^2 72^\circ} \]

This gives us:

\[ 2 + 1 = 2 \]

### Step 3: Final Calculation
Now we can substitute the simplified numerator and denominator back into the expression:

\[ \frac{\sqrt{3}}{2} \]

### Conclusion
Thus, the value of the expression is:

\[ \frac{\sqrt{3}}{2} \]

The correct option is (b) \(\frac{\sqrt{3}}{2}\).