Check the solution to "How to simplify cos⁡(π2−x)⋅csc⁡(−x)?"

Anabella Gilbert

Answered question

2023-01-20

How to simplify

\cos (\frac{π}{2} - x) \cdot \csc (- x)

mosellasv2

Beginner2023-01-21Added 7 answers

We are aware that a right triangle's angles must add up to

π

radians. Since the right angle is

\frac{π}{2}

radians, the other two angles must sum to

\frac{π}{2}

radians. Therefore

\frac{π}{2}

minus the adjacent angle must be the opposite angle. Another way of saying this is

\cos (\frac{π}{2} - x) = \sin x

, so

\cos (\frac{π}{2} - x) \cdot \csc (- x) = \sin x \cdot \csc (- x)

.
The definition of

\csc x

says that

\csc x = \frac{1}{\sin x}

, so

\sin x \cdot \csc (- x) = \frac{\sin x}{\sin (- x)}

We know that

\sin x

is an odd function so

\sin (- x) = - \sin x

and we can write

\frac{\sin x}{\sin (- x)} = - \frac{\sin x}{\sin x} = - 1

.
However, this breaks down when

x

is a multiple of

π

because

\csc (n π)

is not defined.

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