1 + cos56° + cos58° - cos66°. Find the correct option

Trigonometric Identity Problem Solution

Question: Find the value of \(1 + \cos 56^\circ + \cos 58^\circ - \cos 66^\circ\)

Options:

A) \(2 \cos 28^\circ \cos 29^\circ \cos 33^\circ\)

B) \(4 \cos 28^\circ \cos 29^\circ \sin 33^\circ\)

C) \(4 \cos 28^\circ \cos 29^\circ \cos 33^\circ\)

D) \(2 \cos 28^\circ \cos 29^\circ \sin 33^\circ\)

Step-by-Step Solution

Key Formula: \(\cos A + \cos B = 2 \cos\left(\frac{A+B}{2}\right) \cos\left(\frac{A-B}{2}\right)\)

Step 1: Group the terms:

\[ (1 + \cos 56^\circ) + (\cos 58^\circ - \cos 66^\circ) \]

Step 2: Apply sum-to-product identities:

\[ 1 + \cos 56^\circ = 2 \cos^2 28^\circ \quad \text{(Using } 1 + \cos θ = 2 \cos^2(\frac{θ}{2})\text{)} \] \[ \cos 58^\circ - \cos 66^\circ = 2 \sin\left(\frac{66^\circ + 58^\circ}{2}\right) \sin\left(\frac{66^\circ - 58^\circ}{2}\right) \] \[ = 2 \sin 62^\circ \sin 4^\circ = 2 \cos 28^\circ \sin 4^\circ \]

Step 3: Combine results:

\[ 2 \cos^2 28^\circ + 2 \cos 28^\circ \sin 4^\circ = 2 \cos 28^\circ (\cos 28^\circ + \sin 4^\circ) \]

Step 4: Convert \(\cos 28^\circ + \sin 4^\circ\) to product form:

\[ \cos 28^\circ + \cos 86^\circ = 2 \cos 57^\circ \cos 29^\circ = 2 \sin 33^\circ \cos 29^\circ \]

Final Result:

\[ 2 \cos 28^\circ \times 2 \sin 33^\circ \cos 29^\circ = 4 \cos 28^\circ \cos 29^\circ \sin 33^\circ \]

✓ Correct Answer: Option B