is $sin(a)=sqrt{1-cos^2(a)}$ derived from $sin^2(a)+cos^2(a)=1$

I have a problem of $cos(a)=t for $270<=a<=360$

$a)cos(90+a)$
for obvious , my answer is the answer is
$cos(90)cos(a)-sin(90)sin(a)$ and left me with
$=-sin(a)$

But what is the proof $sin(a)=sqrt{(1-cos^2(a)}$
$sin^2(a)+cos^2(a)=1$
$sin^2(a)=1-cos^2(a)$
$sin(a)=sqrt{1-cos^2(a)}$
my friend said that , this is false
What is the real solution or proof to the $sin(a)=sqrt{(1-cos^2(a)}$?