# How do I simplify this expression? Sin²(-x)-Sin(-x)/1-sin(-x)

Mar 20, 2018

$\sin \left(x\right)$

#### Explanation:

Inspection of

(sin^2(-x) - sin(-x))/(1 - sin(-x)

reveals that the numerator may be factorised by extracting the term $\sin \left(- x\right)$.

That is, the expression may be written

(sin(-x) (sin(-x) - 1))/(1 - sin(-x)

$\sin \left(- x\right)$ is an example of an odd function, for which $f \left(- x\right) = - f \left(x\right)$ (which means it has rotational symmetry around the origin, compared with even functions, for which $f \left(- x\right) = f \left(x\right)$, which have reflective symmetry in the y axis)

So, substituting $- \sin \left(x\right)$ for the $\sin \left(- x\right)$ in the term outside the brackets (but leaving the others for now for reasons that will become clear)

(sin(-x) (sin(-x) - 1))/(1 - sin(-x)

implies

(- sin(x) (sin(-x) - 1))/(1 - sin(-x)

which in turn implies

$\frac{\sin \left(x\right) \left(1 - \sin \left(- x\right)\right)}{1 - \sin \left(- x\right)}$

The term $\left(1 - \sin \left(- x\right)\right)$ is a factor of both numerator and denominator so it may be cancelled to leave

$\sin \left(x\right)$