Question

If \( \sin \theta + \csc \theta = 2 \), then the value of \( \sin^6 \theta + \csc^6 \theta \) is equal to:

• A. \( 0 \)
• B. \( 1 \)
• C. \( 2 \)
• D. \( 2^3 \)
• E. \( 2^6 \)

Hint:

For any expression of the form \( x + \frac{1}{x} = 2 \), then \( x^n + \frac{1}{x^n} \) will always be equal to 2 for any integer \( n \).

Answer 1

The Correct Option is C

Concept: Note that \( \csc \theta = \frac{1}{\sin \theta} \). If \( x + \frac{1}{x} = 2 \), the only real solution is \( x = 1 \).

Step 1: Solve for \( \sin \theta \).
Let \( \sin \theta = x \). \[ x + \frac{1}{x} = 2 \] \[ x^2 + 1 = 2x \quad \Rightarrow \quad x^2 - 2x + 1 = 0 \] \[ (x - 1)^2 = 0 \quad \Rightarrow \quad x = 1 \] So, \( \sin \theta = 1 \).

Step 2: Find \( \csc \theta \).
\[ \csc \theta = \frac{1}{\sin \theta} = \frac{1}{1} = 1 \]

Step 3: Calculate the required expression.
\[ \sin^6 \theta + \csc^6 \theta = (1)^6 + (1)^6 \] \[ 1 + 1 = 2 \]