If `tan ^(2) theta + cot^(2) theta = 2 `, then what is the value of `2^(sec theta cosec theta )` ?

Options

A. 0
B. 1
C. 2
D. 4

Answer

To solve the problem step by step, we start with the given equation:

**Step 1: Understand the given equation.**
We have:
\[ \tan^2 \theta + \cot^2 \theta = 2 \]

**Step 2: Use the identity for tangent and cotangent.**
Recall that:
\[ \tan \theta = \frac{\sin \theta}{\cos \theta} \]
\[ \cot \theta = \frac{\cos \theta}{\sin \theta} \]

**Step 3: Rewrite the equation in terms of sine and cosine.**
Using the identities:
\[ \tan^2 \theta = \frac{\sin^2 \theta}{\cos^2 \theta} \]
\[ \cot^2 \theta = \frac{\cos^2 \theta}{\sin^2 \theta} \]
The equation becomes:
\[ \frac{\sin^2 \theta}{\cos^2 \theta} + \frac{\cos^2 \theta}{\sin^2 \theta} = 2 \]

**Step 4: Let \( x = \tan^2 \theta \).**
Then, we can rewrite the equation as:
\[ x + \frac{1}{x} = 2 \]

**Step 5: Solve the equation.**
Multiplying through by \( x \) gives:
\[ x^2 - 2x + 1 = 0 \]
Factoring, we find:
\[ (x - 1)^2 = 0 \]
Thus, \( x = 1 \) which means:
\[ \tan^2 \theta = 1 \]
This implies:
\[ \tan \theta = 1 \]
Therefore, \( \theta = 45^\circ \) (or \( \frac{\pi}{4} \) radians).

**Step 6: Find secant and cosecant values.**
Now, we need to find \( \sec \theta \) and \( \csc \theta \):
- \( \sec 45^\circ = \frac{1}{\cos 45^\circ} = \frac{1}{\frac{1}{\sqrt{2}}} = \sqrt{2} \)
- \( \csc 45^\circ = \frac{1}{\sin 45^\circ} = \frac{1}{\frac{1}{\sqrt{2}}} = \sqrt{2} \)

**Step 7: Calculate \( 2^{\sec \theta \csc \theta} \).**
Now we calculate:
\[ \sec \theta \csc \theta = \sqrt{2} \cdot \sqrt{2} = 2 \]
Thus:
\[ 2^{\sec \theta \csc \theta} = 2^2 = 4 \]

**Final Answer:**
The value of \( 2^{\sec \theta \csc \theta} \) is \( 4 \).

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