If `cosec^(2)theta+cot^(2)theta=7` , 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 identity for cosecant and cotangent
We know that:
\[ \csc^2 \theta = 1 + \cot^2 \theta \]
Substituting this identity into the given equation:
\[ 1 + \cot^2 \theta + \cot^2 \theta = 7 \]

### Step 2: Simplify the equation
Combine like terms:
\[ 1 + 2\cot^2 \theta = 7 \]
Now, subtract 1 from both sides:
\[ 2\cot^2 \theta = 6 \]

### Step 3: Solve for cotangent
Divide both sides by 2:
\[ \cot^2 \theta = 3 \]
Now, take the square root of both sides:
\[ \cot \theta = \sqrt{3} \quad \text{or} \quad \cot \theta = -\sqrt{3} \]

### Step 4: Find the angles
The value \( \cot \theta = \sqrt{3} \) corresponds to:
\[ \theta = 30^\circ \]
The value \( \cot \theta = -\sqrt{3} \) corresponds to:
\[ \theta = 210^\circ \quad (\text{in the third quadrant}) \]

### Step 5: Conclusion
Thus, the possible values of \( \theta \) are \( 30^\circ \) and \( 210^\circ \). However, if we are looking for the principal value, we take:
\[ \theta = 30^\circ \]

### Final Answer:
The value of \( \theta \) is \( 30^\circ \).

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