`tan^(2)theta=1-e^(2)`, then `sectheta+tan^(3)theta.cosectheta=?`

Options

A. `(e^(2)-1)^(3//2)`
B. `1-e`
C. `(2-e^(2))^(3//2)`
D. `(e^(2)-2)^(1//2)`

Answer

To solve the equation \( \tan^2 \theta = 1 - e^2 \) and find the value of \( \sec \theta + \tan^3 \theta \cdot \csc \theta \), we can follow these steps:

### Step 1: Rewrite the expression
We start with the expression we need to evaluate:
\[ \sec \theta + \tan^3 \theta \cdot \csc \theta \]

### Step 2: Substitute identities
Using the trigonometric identities, we know:
- \( \sec \theta = \frac{1}{\cos \theta} \)
- \( \tan \theta = \frac{\sin \theta}{\cos \theta} \)
- \( \csc \theta = \frac{1}{\sin \theta} \)

Thus, we can rewrite \( \tan^3 \theta \cdot \csc \theta \) as:
\[ \tan^3 \theta \cdot \csc \theta = \left(\frac{\sin \theta}{\cos \theta}\right)^3 \cdot \frac{1}{\sin \theta} = \frac{\sin^2 \theta}{\cos^3 \theta} \]

### Step 3: Combine the terms
Now we can combine the terms:
\[ \sec \theta + \tan^3 \theta \cdot \csc \theta = \frac{1}{\cos \theta} + \frac{\sin^2 \theta}{\cos^3 \theta} \]

### Step 4: Find a common denominator
The common denominator for the two fractions is \( \cos^3 \theta \):
\[ \sec \theta + \tan^3 \theta \cdot \csc \theta = \frac{\cos^2 \theta}{\cos^3 \theta} + \frac{\sin^2 \theta}{\cos^3 \theta} = \frac{\cos^2 \theta + \sin^2 \theta}{\cos^3 \theta} \]

### Step 5: Use the Pythagorean identity
Using the Pythagorean identity \( \sin^2 \theta + \cos^2 \theta = 1 \):
\[ \sec \theta + \tan^3 \theta \cdot \csc \theta = \frac{1}{\cos^3 \theta} \]

### Step 6: Express in terms of secant
Since \( \sec \theta = \frac{1}{\cos \theta} \), we can express \( \frac{1}{\cos^3 \theta} \) as:
\[ \sec^3 \theta \]

### Step 7: Find \( \sec \theta \)
From the original equation \( \tan^2 \theta = 1 - e^2 \), we can express \( \sec^2 \theta \):
\[ \sec^2 \theta = 1 + \tan^2 \theta = 1 + (1 - e^2) = 2 - e^2 \]

### Step 8: Calculate \( \sec \theta \)
Taking the square root gives us:
\[ \sec \theta = \sqrt{2 - e^2} \]

### Step 9: Substitute back to find the final answer
Now, substituting back into our expression for \( \sec^3 \theta \):
\[ \sec^3 \theta = \left(\sqrt{2 - e^2}\right)^3 = (2 - e^2)^{3/2} \]

### Final Answer
Thus, the final answer is:
\[ \sec \theta + \tan^3 \theta \cdot \csc \theta = (2 - e^2)^{3/2} \]