If `"cosec"^(2) theta + cot^(2) theta = 7`, then what is the value (in degrees) of `theta`?

Options

A. 15
B. 30
C. 45
D. 60

Answer

To solve the equation \( \csc^2 \theta + \cot^2 \theta = 7 \), we can follow these steps:

### Step 1: Use the Pythagorean identity
We know that:
\[ \csc^2 \theta = 1 + \cot^2 \theta \]
Substituting this into the original equation gives:
\[ 1 + \cot^2 \theta + \cot^2 \theta = 7 \]

### Step 2: Simplify the equation
Combine the cotangent terms:
\[ 1 + 2\cot^2 \theta = 7 \]

### Step 3: Isolate the cotangent term
Subtract 1 from both sides:
\[ 2\cot^2 \theta = 7 - 1 \]
\[ 2\cot^2 \theta = 6 \]

### Step 4: Solve for \( \cot^2 \theta \)
Divide both sides by 2:
\[ \cot^2 \theta = \frac{6}{2} \]
\[ \cot^2 \theta = 3 \]

### Step 5: Take the square root
Taking the square root of both sides gives:
\[ \cot \theta = \sqrt{3} \]

### Step 6: Find the angle \( \theta \)
The value of \( \theta \) for which \( \cot \theta = \sqrt{3} \) is:
\[ \theta = 30^\circ \]

Thus, the value of \( \theta \) is \( 30^\circ \).

### Final Answer
\[ \theta = 30^\circ \]

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