We know that,
x^circ=((pix)/180)^R=(pix)/180
Let,
f(x)=sin((pix)/180)=>f(t)=sin((pit)/180)
Now,
color(blue)(f'(x)=lim_(t tox) (f(t)-f(x))/(t-x)
color(white)(f'(x))=lim_(t tox)(sin((pit)/180)-sin((pix)/180))/(t-x)
color(white)(f'(x))=lim_(t tox)(2cos(((pit)/180+(pix)/180)/2)sin(((pit)/180-(pix)/180)/2))/(t-x)
color(white)(f'(x))=lim_(t tox)(2cos(pi/360(t+x))sin((pi/360(t-x))))/((pi/360(t-x)))*pi/360
color(white)(f'(x))=(2pi)/360lim_(t tox)cos(pi/360(t+x))*lim_(t tox)[sin(pi/360(t-x))/(pi/360(t-x))]
Now,
t tox=>(t-x)to0=>pi/360(t-x)to0 and lim_(thetato0)sintheta/theta=1
:.f'(x)=pi/180cos(pi/360(x+x))*(1)
f'(x)=pi/180cos(pi/360(2x))
f'(x)=pi/180cos((pix)/180)^R , where ,(pix)/180=x^circ
f'(x)=pi/180cosx^circ