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#