First principle of differentiation :\\\\(\\dfrac {dy} {dx} =\\lim_{\\delta x\\rightarrow 0} \\dfrac {f(x+\\delta x) - f(x)} {\\delta x} \\\\)Here \\\\(f(x) =\\sin x \\\\)\\\\(\\Rightarrow f(x+\\delta x) =\\sin(x+\\delta x) \\\\)\\\\(\\Rightarrow f(x+\\delta x) - f(x) =\\sin(x+\\delta x) - \\sin x \\\\)We know that \\\\(\ ext {sin} C-\ ext {sin} D=2\ ext {cos} (\\dfrac {C+D} {2})\ ext {sin} (\\dfrac {C-D} {2})\\\\)\\\\(\\Rightarrow f(x+\\delta x) - f(x) =2\\cos(\\dfrac {x+\\delta x+x}{2})\\sin(\\dfrac {\\delta x} {2})\\\\)\\\\(\\Rightarrow \\dfrac {dy} {dx} =\\lim _{\\delta x\\rightarrow 0} \\dfrac {2\\cos(x+\\dfrac {\\delta x} {2})\\sin(\\dfrac {\\delta x} {2})}{\\delta x} \\\\)\\\\(\\Rightarrow \\dfrac {dy} {dx} =\\lim _{\\delta x \\rightarrow 0} \\cos(x+\\dfrac {\\delta x} {2})\\dfrac {\\sin(\\dfrac {\\delta x} {2})}{\\dfrac {\\delta x} {2}}\\\\)\\\\(\\Rightarrow \\dfrac {d(\\sin x)} {dx} =\\cos x\\\\) as \\\\(\\lim_{x\\rightarrow 0} \\dfrac {\\sin x} {x} =1\\\\)