T-Ratios for Angle of 45°

Let a ray OA start from OX and rotate in the anti-clockwise direction and make an angle of 45° with the x-axis.

Take any point P on OA. Draw PM ⊥ OX.

Now in right ΔPMO, 

∠POM + ∠OPM + ∠PMO = 180°

45° + ∠OPM + 90° = 180°

∠OPM = 180° - 90° - 45° = 45°

In ΔPMO, ∠OPM = ∠POM = 45°

OM = PM = a units

In right triangle PMO,

OP² = OM² + PM² (Pythagoras Theorem)

= a² + a²

= 2a²

$$ OP = \sqrt{2}a $$

$$ \sin 45° = \frac{1}{\sqrt{2}} $$

$$ \cos 45° = \frac{1}{\sqrt{2}} $$

$$ \tan 45° = 1 $$

$$ \csc 45° = \sqrt{2} $$

$$ \sec 45° = \sqrt{2} $$

$$ \cot 45° = 1 $$