What is the value of `( cosec (78^(@) - theta) - sec ( 12^(@) + theta) - tan (67^(@) + theta) + cot (23^(@) - theta) )/( tan 13^(@) tan 37^(@) tan 45^(@) tan 53^(@) tan 77^(@) )`?

Options

A. `-1`
B. 2
C. 1
D. 0

Answer

To solve the given expression step-by-step, we will simplify the numerator and the denominator separately.

### Given Expression:
\[ \frac{ \csc(78^\circ - \theta) - \sec(12^\circ + \theta) - \tan(67^\circ + \theta) + \cot(23^\circ - \theta) }{ \tan(13^\circ) \tan(37^\circ) \tan(45^\circ) \tan(53^\circ) \tan(77^\circ) } \]

### Step 1: Simplifying the Numerator
1. **Rewrite Trigonometric Functions**:
- Using the identity \(\csc(90^\circ - x) = \sec(x)\):
\[ \csc(78^\circ - \theta) = \sec(12^\circ + \theta) \]
- Using the identity \(\tan(90^\circ - x) = \cot(x)\):
\[ \tan(67^\circ + \theta) = \cot(23^\circ - \theta) \]

2. **Substituting the Identities**:
- Substitute these into the numerator:
\[ \sec(12^\circ + \theta) - \sec(12^\circ + \theta) - \tan(67^\circ + \theta) + \cot(23^\circ - \theta) \]
- This simplifies to:
\[ 0 - \tan(67^\circ + \theta) + \cot(23^\circ - \theta) \]

3. **Further Simplification**:
- Since \(\tan(67^\circ + \theta) = \cot(23^\circ - \theta)\), we can write:
\[ -\tan(67^\circ + \theta) + \tan(67^\circ + \theta) = 0 \]

### Step 2: Simplifying the Denominator
1. **Evaluate the Denominator**:
- The denominator is:
\[ \tan(13^\circ) \tan(37^\circ) \tan(45^\circ) \tan(53^\circ) \tan(77^\circ) \]
- We know that \(\tan(45^\circ) = 1\), so:
\[ \tan(13^\circ) \tan(37^\circ) \cdot 1 \cdot \tan(53^\circ) \tan(77^\circ) \]

2. **Using Complementary Angles**:
- Notice that:
\[ \tan(53^\circ) = \cot(37^\circ) \quad \text{and} \quad \tan(77^\circ) = \cot(13^\circ) \]
- Therefore:
\[ \tan(13^\circ) \tan(77^\circ) = 1 \quad \text{and} \quad \tan(37^\circ) \tan(53^\circ) = 1 \]
- Thus, the denominator simplifies to:
\[ 1 \cdot 1 = 1 \]

### Final Step: Combine Numerator and Denominator
- The entire expression simplifies to:
\[ \frac{0}{1} = 0 \]

### Final Answer:
The value of the expression is:
\[ \boxed{0} \]