Here clearly both \\\\(\\sin x\\\\) and \\\\(\\cos x\\\\) are defined in their domain.Let's assume that \\\\(g(x)=\\sin x\\\\) and \\\\(f(x)=\\cos x\\\\)=> Let's first prove that \\\\(g(x)\\\\) is continuous in it's domain.Let \\\\(c\\\\) be a real number, put \\\\(x=c+h\\\\) So if \\\\(x \\Rightarrow c\\\\) then it means that \\\\(h \\Rightarrow 0\\\\)\\\\(\\underset { x\\Rightarrow c }{ \\lim } \\\\) \\\\(g(x)\\\\) = \\\\(\\underset { x\\Rightarrow c }{ \\lim } \\\\) \\\\(sin(x)\\\\)Put \\\\(x=h+c\\\\)And as mentioned above, when \\\\(x \\Rightarrow c\\\\) then it means that \\\\(h \\Rightarrow 0\\\\)Which gives us \\\\(\\underset { h\\Rightarrow 0 }{ \\lim } \\\\) \\\\(\\sin(c+h)\\\\)Expanding \\\\(\\sin(c+h)\\\\) = \\\\(\\sin(h)\\cos(c)+\\cos(h)\\sin(c)\\\\)Which gives us \\\\(\\underset { h\\Rightarrow 0 }{ \\lim } \\\\) \\\\(\\sin(h)\\cos(c)+\\cos(h)\\sin(c)\\\\) \\\\(=\\sin(c)\\cos(0)+\\cos(c)\\sin(0)\\\\) \\\\(=\\sin(c)\\\\)So here we get, \\\\(\\underset { x\\rightarrow c }{ \\lim } \\\\) \\\\(g(x)\\\\) = \\\\(\\underset { x\\Rightarrow c }{ \\lim } \\\\) \\\\(\\sin(x)\\\\) = \\\\(\\sin(c) = g(c)\\\\)And this proves that \\\\(\\sin(x)\\\\) is continuous all across its domain=> Let's prove that \\\\(f(x)=\\cos(x)\\\\) is continuous in it's domain.Let \\\\(c\\\\) be a real number, put \\\\(x=c+h\\\\) So if \\\\(x \\Rightarrow c\\\\) then it means that \\\\(h \\Rightarrow 0\\\\)\\\\(f(c)= \\cos(c)\\\\)\\\\(\\underset { x\\Rightarrow c }{ \\lim } \\\\) \\\\(f(x)\\\\) = \\\\(\\underset { x\\Rightarrow c }{ \\lim } \\\\) \\\\(\\cos(x)\\\\)Put \\\\(x=h+c\\\\)And as mentioned above, when \\\\(x \\Rightarrow c\\\\) then it means that \\\\(h \\Rightarrow 0\\\\)Which gives us \\\\(\\underset { h\\Rightarrow 0 }{ \\lim } \\\\) \\\\(\\cos(c+h)\\\\)Expanding \\\\(\\cos(h+c)\\\\) = \\\\(\\cos(h)\\cos(c)-\\sin(h)\\sin(c)\\\\)Which gives us \\\\(\\underset { h\\Rightarrow 0 }{ \\lim } \\\\) \\\\(\\cos(h)\\cos(c)-\\sin(h)\\sin(c)\\\\) \\\\(=\\cos(c)\\cos(0)-\\sin(c)\\sin(0)\\\\) \\\\(=\\cos(c)\\\\)This gives us\\\\(\\underset { x\\Rightarrow c }{ \\lim } \\\\) \\\\(f(x)\\\\) = \\\\(\\underset { x\\Rightarrow c }{ \\lim } \\\\) \\\\(\\cos(x)\\\\) = \\\\(\\cos(c) = f(c)\\\\)And this proves that \\\\(\\cos(x)\\\\) is continuous all across its domain=> So by theorem, if function \\\\(f\\\\) and function \\\\(g\\\\) are continous, then \\\\(f.g\\\\) is also continous.There for \\\\(\\sin(x).\\cos(x)\\\\) is continous.