Complete the identity.
-$\frac { \cos ( 3 \theta ) - \cos ( 9 \theta ) } { \cos ( 3 \theta ) + \cos ( 9 \theta ) } = ?$
A) 0
B) $\tan ( 3 \theta ) \tan ( 6 \theta )$
C) $\cot ( 6 \theta )$
D) $- \tan ( 6 \theta )$

Question

Quizplus · Accepted Answer

The given expression can be simplified using the trigonometric identity for the tangent of the sum of angles, specifically using the formula for converting sums and differences of cosines into products. The identity $\frac{\cos A - \cos B}{\cos A + \cos B} = -\tan\left(\frac{A - B}{2}\right)\tan\left(\frac{A + B}{2}\right)$ applies here, with $A = 3\theta$ and $B = 9\theta$. Substituting these values gives $\tan(3\theta)\tan(6\theta)$, which matches choice B.

Complete the Identity cos⁡(3θ)−cos⁡(9θ)cos⁡(3θ)+cos⁡(9θ)=?\frac { \cos ( 3 \theta ) - \cos ( 9 \theta ) } { \cos ( 3 \theta ) + \cos ( 9 \theta ) } = ?cos(3θ)+cos(9θ)cos(3θ)−cos(9θ)​=?

Complete the Identity $\frac { \cos ( 3 \theta ) - \cos ( 9 \theta ) } { \cos ( 3 \theta ) + \cos ( 9 \theta ) } = ?$