What is the value of
cosec `(65^(@) + theta) - sec(25^(@) - theta) + tan^(2) 20^(@) - cosec^(2) 70^(@)` ?

Options

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

Answer

To solve the expression \( \csc(65^\circ + \theta) - \sec(25^\circ - \theta) + \tan^2(20^\circ) - \csc^2(70^\circ) \), we can follow these steps:

### Step 1: Rewrite the cosecant and secant functions
We know that:
\[ \csc(x) = \frac{1}{\sin(x)} \quad \text{and} \quad \sec(x) = \frac{1}{\cos(x)} \]
Thus, we can rewrite the expression as:
\[ \frac{1}{\sin(65^\circ + \theta)} - \frac{1}{\cos(25^\circ - \theta)} + \tan^2(20^\circ) - \csc^2(70^\circ) \]

### Step 2: Use trigonometric identities
Using the identity \( \sin(90^\circ - x) = \cos(x) \), we can express:
\[ \sin(65^\circ + \theta) = \cos(25^\circ - \theta) \]
This means:
\[ \csc(65^\circ + \theta) = \sec(25^\circ - \theta) \]

### Step 3: Substitute the identities into the expression
Now substituting back into the expression, we have:
\[ \sec(25^\circ - \theta) - \sec(25^\circ - \theta) + \tan^2(20^\circ) - \csc^2(70^\circ) \]

### Step 4: Simplify the expression
The first two terms cancel each other out:
\[ 0 + \tan^2(20^\circ) - \csc^2(70^\circ) \]
Now, we know that:
\[ \csc^2(70^\circ) = 1 + \cot^2(70^\circ) \]
And since \( \cot(70^\circ) = \tan(20^\circ) \), we can write:
\[ \csc^2(70^\circ) = 1 + \tan^2(20^\circ) \]

### Step 5: Substitute and simplify further
Now substituting this back into our expression gives:
\[ \tan^2(20^\circ) - (1 + \tan^2(20^\circ)) \]
This simplifies to:
\[ \tan^2(20^\circ) - 1 - \tan^2(20^\circ) = -1 \]

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