QUESTION

The trigonometric equation $\cos 4x\cdot \cos 8x-\cos 5x\cdot \cos 9x=0$ if This question has multiple correct options  (a) $\cos (12x)=\cos (14x)$   (b) $\sin (13x)=0$   (C) $\sin x=0$   (d) $\cos x=0$

Hint: You can use the product-to-sum and sum-to-product formulas to rewrite the product and sum of the cosine, respectively for expanding or simplifying given trigonometric expressions.

Complete step-by-step solution -
The given trigonometric equation can be written as
$\cos 4x\cdot \cos 8x-\cos 5x\cdot \cos 9x=0$
$\cos 4x\cdot \cos 8x=\cos 5x\cdot \cos 9x$
Multiplying both sides by 2, we get
$2\cos 4x\cdot \cos 8x=2\cos 5x\cdot \cos 9x$
Applying the formula for the product of cosines $2\cos A\cos B=\cos (A-B)+\cos (A+B)$ , we get
We can then substitute the given angles into the formula and simplify.
$\cos \left( 4x-8x \right)+\cos \left( 4x+8x \right)=\cos \left( 5x-9x \right)+\cos \left( 5x+9x \right)$
$\cos \left( -4x \right)+\cos \left( 12x \right)=\cos \left( -4x \right)+\cos \left( 14x \right)$
We know that, $\cos (-\theta )=\cos \theta$
$\cos \left( 4x \right)+\cos \left( 12x \right)=\cos \left( 4x \right)+\cos \left( 14x \right)$
Cancelling the term $\cos (4x)$ on both sides, we get
$\cos \left( 12x \right)=\cos \left( 14x \right)............(1)$
Hence the correct option for the given trigonometric equation is option (a).
The equation (1) can be written as
$\cos \left( 12x \right)-\cos \left( 14x \right)=0$
Applying the formula for the sum of the cosine $\cos A-\cos B=-2\sin \left( \dfrac{A+B}{2} \right)sin\left( \dfrac{A-B}{2} \right)$ , we get
We can then substitute the given angles into the formula and simplify.
$-2\sin \left( \dfrac{12x+14x}{2} \right)sin\left( \dfrac{12x-14x}{2} \right)=0$
$-2\sin \left( \dfrac{26x}{2} \right)sin\left( \dfrac{-2x}{2} \right)=0$
$-2\sin \left( 13x \right)sin\left( -x \right)=0$
We know that $\sin (-\theta )=-\sin \theta$
$2\sin \left( 13x \right)sin\left( x \right)=0$
Dividing both sides by 2, we get
$\sin \left( 13x \right)sin\left( x \right)=0$
$\sin \left( 13x \right)\text{=0 or }sin\left( x \right)=0$
Hence the correct options of the given trigonometric equation are option (b) and option(c).
Therefore, the correct options of the given question are option (a), option (b) and option(c).

Note: It is not true that cos (A) cos (B) is equal to cos (AB). There is no nice formula for cos (AB). You can use the product to sum formulas of the cosine for the trigonometric expression cos (A) cos (B).