What is the value of `(tan theta cosec theta)^(2)- (sin theta sec theta)^(2)`.

Options

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

Answer

To find the value of \((\tan \theta \csc \theta)^2 - (\sin \theta \sec \theta)^2\), we will simplify the expression step by step.

### Step 1: Rewrite the trigonometric functions
We start by rewriting \(\tan \theta\), \(\csc \theta\), \(\sin \theta\), and \(\sec \theta\) 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 the trigonometric identities
Now, we substitute these identities into the 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 rewrite the original expression using these substitutions:
\[ (\tan \theta \csc \theta)^2 - (\sin \theta \sec \theta)^2 = \sec^2 \theta - \tan^2 \theta \]

### Step 4: Apply the Pythagorean identity
We use the Pythagorean identity:
\[ \sec^2 \theta - \tan^2 \theta = 1 \]

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