Complete the identity.
-$$\frac { \sin ( 4 \theta ) + \sin ( 8 \theta ) } { \cos ( 4 \theta ) + \cos ( 8 \theta ) } = ?$$
A) $\tan ( 4 \theta ) + \tan ( 8 \theta )$
B) $\tan ( 6 \theta )$
C) $\tan ( 6 \theta ) \cot ( 2 \theta )$
D) $2 \tan ( 6 \theta ) \tan ( 2 \theta )$

Question

Quizplus · Accepted Answer

The given expression can be simplified using the tangent addition formula. The formula for tangent of sum of two angles, $\tan(A + B) = \frac{\sin(A + B)}{\cos(A + B)}$, can be applied here by setting $A = 4\theta$ and $B = 8\theta$, leading to $\tan(4\theta + 8\theta) = \tan(12\theta)$, which simplifies to $\tan(6\theta)$ due to the periodicity and symmetry properties of the tangent function.

Complete the Identity sin⁡(4θ)+sin⁡(8θ)cos⁡(4θ)+cos⁡(8θ)=?\frac { \sin ( 4 \theta ) + \sin ( 8 \theta ) } { \cos ( 4 \theta ) + \cos ( 8 \theta ) } = ?cos(4θ)+cos(8θ)sin(4θ)+sin(8θ)​=?

Complete the Identity $\frac { \sin ( 4 \theta ) + \sin ( 8 \theta ) } { \cos ( 4 \theta ) + \cos ( 8 \theta ) } = ?$