Question

If \(a_n = \cos 2n\), then \(a_1 \times a_2 \times \dots \times a_{45} =\)

• A. \(\frac{\sqrt3}{2}\)
• B. \(\frac12\)
• C. \(0\)
• D. \(1\)

Hint:

In long products involving trig terms, always check if any term becomes zero before doing full computation.

Answer 1

The Correct Option is C

Concept: If any term in a product is zero, the whole product becomes zero.

Step 1: Check if any term becomes zero. \[ a_n=\cos(2n) \] We check if: \[ \cos(2n)=0 \] \[ 2n=90^\circ, 270^\circ, \dots \] \[ n=45^\circ \] So: \[ a_{45}=\cos(90^\circ)=0 \]

Step 2: Conclude product. Since one term is zero: \[ a_1 a_2 \cdots a_{45}=0 \] \[ \boxed{0} \]