Question

Cos 4x cos 8x - cos 5x cos 9x = 0 if

• A. Cos 12x = cos 14x
• B. Sin 13x = 0
• C. Sin x = 0
• D. Cos x = 0

Answer 1

Answer: (A), (B), (C)

Sin x = 0

We are not able to guess/find the solutions directly from the given expression. We have to

modify/simplify the expression to guess the solution. We can apply transformation formula for cos 4x cos 8x and cos 5x cos 9x.
⇒ $\frac{1}{2}$ (cos 12x + cos 4x - cos 4x - cos 14x) = 0
⇒ cos 12x = cos 14x ⇒ (A)
⇒ 12x = 2nπ ± 14x
We will split this into

12x = 2nπ + 14x or 12x = 2nπ - 14x

⇒x= -nπ or 13x = nπ
Checking the other options
B is true because we have 13x = nπ

⇒sin x = sin(- nπ) = 0
C is also true because we have x = -nπ

⇒ sin x = sin (-nπ ) = 0
D is not true because

cos x = cos (-nπ ) $\ne$ 0