Use the half-angle formulas to simplify the expression. $$- \sqrt { \frac { 1 + \cos 6 x } { 1 - \cos 6 x } }$$
A)- |cot x|
B)- |3 tan x|
C)- |3 tan 6x|
D)- |3 cot 3x|
E)- |cot 3x|

Question

Use the half-angle formulas to simplify the expression.​ $$- \sqrt { \frac { 1 + \cos 6 x } { 1 - \cos 6 x } }$$ ​
A)​- |cot x|
B)​- |3 tan x|
C)​- |3 tan 6x|
D)​- |3 cot 3x|
E)​- |cot 3x|

Quizplus · Accepted Answer

The given expression can be simplified using the half-angle formula for tangent, which is $\tan\left(\frac{\theta}{2}\right) = \sqrt{\frac{1 - \cos\theta}{1 + \cos\theta}}$. By applying this to the given expression and considering the negative square root, we get $-\sqrt{\frac{1 + \cos6x}{1 - \cos6x}}$, which simplifies to $-|\cot3x|$ by recognizing that the expression inside the square root is the reciprocal of the half-angle formula for tangent, thus leading to the cotangent of half the angle (3x in this case).

Use the Half-Angle Formulas to Simplify the Expression −1+cos⁡6x1−cos⁡6x- \sqrt { \frac { 1 + \cos 6 x } { 1 - \cos 6 x } }−1−cos6x1+cos6x​​

Use the Half-Angle Formulas to Simplify the Expression $- \sqrt { \frac { 1 + \cos 6 x } { 1 - \cos 6 x } }$