What is the value of `(tan thetacosectheta)^2 - (sintheta sec theta)^2` ?

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:
- \(\tan \theta = \frac{\sin \theta}{\cos \theta}\)
- \(\csc \theta = \frac{1}{\sin \theta}\)
- \(\sec \theta = \frac{1}{\cos \theta}\)

Thus, we can express \((\tan \theta \csc \theta)^2\) as:
\[ (\tan \theta \csc \theta)^2 = \left(\frac{\sin \theta}{\cos \theta} \cdot \frac{1}{\sin \theta}\right)^2 = \left(\frac{1}{\cos \theta}\right)^2 = \sec^2 \theta \]

### Step 2: Simplify the second term
Next, we simplify \((\sin \theta \sec \theta)^2\):
\[ (\sin \theta \sec \theta)^2 = \left(\sin \theta \cdot \frac{1}{\cos \theta}\right)^2 = \left(\frac{\sin \theta}{\cos \theta}\right)^2 = \tan^2 \theta \]

### Step 3: Substitute back into the expression
Now we substitute these simplified forms back into the original expression:
\[ (\tan \theta \csc \theta)^2 - (\sin \theta \sec \theta)^2 = \sec^2 \theta - \tan^2 \theta \]

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

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