The value of the expression `cosec(75^(@)+theta) - sec(15^@-theta)-tan(55^(@)+theta) + cot(35^(@)-theta)` is

Options

A. `-1`
B. `0`
C. 1
D. `3/2`

Explanation

To solve the expression \( \csc(75^\circ + \theta) - \sec(15^\circ - \theta) - \tan(55^\circ + \theta) + \cot(35^\circ - \theta) \), we can use trigonometric identities and properties. Let's break it down step by step.

### Step-by-Step Solution:

1. **Rewrite the Cosecant**:
\[ \csc(75^\circ + \theta) = \frac{1}{\sin(75^\circ + \theta)} \]

2. **Rewrite the Secant**:
\[ \sec(15^\circ - \theta) = \frac{1}{\cos(15^\circ - \theta)} \]

3. **Rewrite the Tangent**:
\[ \tan(55^\circ + \theta) = \frac{\sin(55^\circ + \theta)}{\cos(55^\circ + \theta)} \]

4. **Rewrite the Cotangent**:
\[ \cot(35^\circ - \theta) = \frac{\cos(35^\circ - \theta)}{\sin(35^\circ - \theta)} \]

5. **Use Trigonometric Identities**:
- We know that \( \sin(75^\circ + \theta) = \cos(15^\circ - \theta) \) (since \( 75^\circ = 90^\circ - 15^\circ \)).
- Also, \( \tan(55^\circ + \theta) = \cot(35^\circ - \theta) \) (since \( 55^\circ = 90^\circ - 35^\circ \)).

6. **Substituting Values**:
- Substitute the identities into the expression:
\[ \csc(75^\circ + \theta) = \sec(15^\circ - \theta) \]
- Therefore, the expression simplifies to:
\[ \sec(15^\circ - \theta) - \sec(15^\circ - \theta) - \tan(55^\circ + \theta) + \tan(55^\circ + \theta) \]

7. **Cancellation**:
- The terms \( \sec(15^\circ - \theta) \) cancel out:
\[ 0 - \tan(55^\circ + \theta) + \tan(55^\circ + \theta) = 0 \]

8. **Final Result**:
- Thus, the value of the expression is:
\[ \boxed{0} \]