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 find the value of the expression \( \csc(75^\circ + \theta) - \sec(15^\circ - \theta) - \tan(55^\circ + \theta) + \cot(35^\circ - \theta) \), we can simplify it step by step using trigonometric identities.

### Step 1: Rewrite the terms using trigonometric identities
1. Recall that \( \csc(x) = \sec(90^\circ - x) \) and \( \cot(x) = \tan(90^\circ - x) \).
2. Rewrite \( \csc(75^\circ + \theta) \) as \( \sec(90^\circ - (75^\circ + \theta)) = \sec(15^\circ - \theta) \).
3. Rewrite \( \cot(35^\circ - \theta) \) as \( \tan(90^\circ - (35^\circ - \theta)) = \tan(55^\circ + \theta) \).

Thus, the expression becomes:
\[ \sec(15^\circ - \theta) - \sec(15^\circ - \theta) - \tan(55^\circ + \theta) + \tan(55^\circ + \theta) \]

### Step 2: Simplify the expression
1. Notice that \( \sec(15^\circ - \theta) - \sec(15^\circ - \theta) = 0 \).
2. Similarly, \( -\tan(55^\circ + \theta) + \tan(55^\circ + \theta) = 0 \).

Putting it all together, we have:
\[ 0 - 0 = 0 \]

### Conclusion
The value of the expression \( \csc(75^\circ + \theta) - \sec(15^\circ - \theta) - \tan(55^\circ + \theta) + \cot(35^\circ - \theta) \) is \( 0 \).

### Final Answer
Therefore, the value of the given trigonometric expression is \( 0 \).

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