Answer:
This result can be verified using a trigonometric identity.
Step-by-step explanation:
We use the the trigonometric identity
\(cos(a\:{\pm}\:b)=cos(a)*cos(b){\mp}sin(a)*sin(b).\)
In our case \(a=x\) and \(b=\pi /2\), thus:
\(cos(x+\frac{\pi}{2} )=cos(x)*cos(\frac{\pi}{2})-sin(x)*sin(\frac{\pi}{2}).\)
Since
\(cos(\frac{\pi}{2})=0\) and
\(sin(\frac{\pi}{2})=1\)
the above equation simplifies as
\(cos(x+\frac{\pi}{2} )=cos(x)*cos(\frac{\pi}{2})-sin(x)*sin(\frac{\pi}{2})=-sin(x)\)
\(\boxed{cos(x+\frac{\pi}{2} )=-sin(x)}\)
thus proving the identity.
