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

1 Answer
Mar 20, 2018

sin(x)

Explanation:

Inspection of

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

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

That is, the expression may be written

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

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

So, substituting -sin(x) for the sin(-x) 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

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

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

sin(x)