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 will use trigonometric identities to simplify it step by step.

### Step 1: Rewrite the cosecant and cotangent terms
Using the identity \( \csc(x) = \sec(90^\circ - x) \) and \( \cot(x) = \tan(90^\circ - x) \), we can rewrite the expression:
\[ \csc(75^\circ + \theta) = \sec(90^\circ - (75^\circ + \theta)) = \sec(15^\circ - \theta) \]
\[ \cot(35^\circ - \theta) = \tan(90^\circ - (35^\circ - \theta)) = \tan(55^\circ + \theta) \]

### Step 2: Substitute the identities into the expression
Now, substituting these identities into the original expression:
\[ \sec(15^\circ - \theta) - \sec(15^\circ - \theta) - \tan(55^\circ + \theta) + \tan(55^\circ + \theta) \]

### Step 3: Simplify the expression
Now we can see that:
\[ \sec(15^\circ - \theta) - \sec(15^\circ - \theta) = 0 \]
\[ -\tan(55^\circ + \theta) + \tan(55^\circ + \theta) = 0 \]
Thus, the entire expression simplifies to:
\[ 0 + 0 = 0 \]

### Final Answer
The value of the expression is \( \boxed{0} \).