Perform the indicated operations and simplify the result so there are no quotients.
-$$\frac { \sin \theta } { 1 + \sin \theta } - \frac { \sin \theta } { 1 - \sin \theta }$$
A) $1 + \cot \theta$
B) $\sec \theta \csc \theta$
C) $- 2 \tan ^ { 2 } \theta$
D) $\sin \theta \tan \theta$

Question

Quizplus · Accepted Answer

To simplify the given expression, we first find a common denominator and combine the fractions:$$\frac { \sin \theta } { 1 + \sin \theta } - \frac { \sin \theta } { 1 - \sin \theta } = \frac { \sin \theta (1 - \sin \theta) - \sin \theta (1 + \sin \theta) } { (1 + \sin \theta)(1 - \sin \theta) }$$Simplifying the numerator and denominator:$$= \frac { \sin \theta - \sin^2 \theta - \sin \theta - \sin^2 \theta } { 1 - \sin^2 \theta }$$$$= \frac { -2\sin^2 \theta } { \cos^2 \theta }$$$$= -2 \left( \frac { \sin^2 \theta } { \cos^2 \theta } \right)$$$$= -2 \tan^2 \theta$$Therefore, the correct answer is $-2 \tan^2 \theta$, which corresponds to choice C.

Perform the Indicated Operations and Simplify the Result So There sin⁡θ1+sin⁡θ−sin⁡θ1−sin⁡θ\frac { \sin \theta } { 1 + \sin \theta } - \frac { \sin \theta } { 1 - \sin \theta }1+sinθsinθ​−1−sinθsinθ​

Perform the Indicated Operations and Simplify the Result So There $\frac { \sin \theta } { 1 + \sin \theta } - \frac { \sin \theta } { 1 - \sin \theta }$