Equation of circle $${x^2} + {y^2} = 1 = {\left( 1 \right)^2}$$

$$ \Rightarrow {x^2} + {y^2} = {\left( {y - mx} \right)^2}$$

$$ \Rightarrow {x^2} = {m^2}{x^2} - 2\,\,mxy;$$

$$ \Rightarrow {x^2}\left( {1 - {m^2}} \right) + 2mxy = 0.$$

Which represents the pair of lines between which the angle is $${45^ \circ }.$$

$$\tan 45 = \pm {{2\sqrt {{m^2} - 0} } \over {1 - {m^2}}} = {{ \pm 2m} \over {1 - {m^2}}};$$

$$ \Rightarrow 1 - {m^2} = \pm 2m$$

$$ \Rightarrow {m^2} \pm 2m - 1 = 0$$

$$ \Rightarrow m = {{ - 2 \pm \sqrt {4 + 4} } \over 2}$$

$$ = {{ - 2 \pm 2\sqrt 2 } \over 2}$$

$$ = - 1 \pm \sqrt 2 .$$