The value of `tan^(2)48^(@)-cosec^(2)42^(@)+cosec(67^(@)+theta)-sec(23^(@)-theta)` is :

Options

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

Answer

To solve the expression \( \tan^2 48^\circ - \csc^2 42^\circ + \csc(67^\circ + \theta) - \sec(23^\circ - \theta) \), we will break it down step by step.

### Step 1: Rewrite the expression using trigonometric identities
We start with the expression:
\[ \tan^2 48^\circ - \csc^2 42^\circ + \csc(67^\circ + \theta) - \sec(23^\circ - \theta) \]

### Step 2: Apply the complementary angle identity
Using the complementary angle identity, we know:
\[ \tan(90^\circ - x) = \cot x \quad \text{and} \quad \csc(90^\circ - x) = \sec x \]
Thus, we can rewrite \( \tan^2 48^\circ \) as:
\[ \tan^2 48^\circ = \cot^2 (90^\circ - 48^\circ) = \cot^2 42^\circ \]

### Step 3: Substitute and simplify
Now substituting \( \tan^2 48^\circ \) in the expression:
\[ \cot^2 42^\circ - \csc^2 42^\circ + \csc(67^\circ + \theta) - \sec(23^\circ - \theta) \]
We know that:
\[ \cot^2 x - \csc^2 x = -1 \]
So, substituting this into our expression gives:
\[ -1 + \csc(67^\circ + \theta) - \sec(23^\circ - \theta) \]

### Step 4: Simplify the remaining terms
Next, we analyze \( \csc(67^\circ + \theta) \) and \( \sec(23^\circ - \theta) \):
Using the identity \( \sec(90^\circ - x) = \csc x \):
\[ \sec(23^\circ - \theta) = \csc(90^\circ - (23^\circ - \theta)) = \csc(67^\circ + \theta) \]
Thus, we have:
\[ \csc(67^\circ + \theta) - \sec(23^\circ - \theta) = \csc(67^\circ + \theta) - \csc(67^\circ + \theta) = 0 \]

### Step 5: Combine the results
Putting it all together, we have:
\[ -1 + 0 = -1 \]

### Final Answer
The value of the expression is:
\[ \boxed{-1} \]