The value of `(tan theta cosec theta)^(2) - (sin theta sec theta)^(2)` is:

Options

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

Answer

To solve the expression \((\tan \theta \csc \theta)^2 - (\sin \theta \sec \theta)^2\), we can follow these steps:

### Step 1: Rewrite the trigonometric functions
We start by rewriting the trigonometric functions in terms of sine and cosine:
\[ \tan \theta = \frac{\sin \theta}{\cos \theta}, \quad \csc \theta = \frac{1}{\sin \theta}, \quad \sec \theta = \frac{1}{\cos \theta} \]

### Step 2: Substitute into the expression
Now, substituting these definitions into the original expression:
\[ (\tan \theta \csc \theta)^2 = \left(\frac{\sin \theta}{\cos \theta} \cdot \frac{1}{\sin \theta}\right)^2 = \left(\frac{1}{\cos \theta}\right)^2 = \sec^2 \theta \]
\[ (\sin \theta \sec \theta)^2 = \left(\sin \theta \cdot \frac{1}{\cos \theta}\right)^2 = \left(\frac{\sin \theta}{\cos \theta}\right)^2 = \tan^2 \theta \]

### Step 3: Substitute back into the expression
Now we can substitute these back into the expression:
\[ \sec^2 \theta - \tan^2 \theta \]

### Step 4: Use the Pythagorean identity
We know from the Pythagorean identity that:
\[ \sec^2 \theta - \tan^2 \theta = 1 \]

### Final Answer
Thus, the value of the expression \((\tan \theta \csc \theta)^2 - (\sin \theta \sec \theta)^2\) is:
\[ \boxed{1} \]