Two angles are complementary if their sum is 90°. If the sum of two angles A and B is 90°, then ∠A and ∠B are complementary angles and each of them is complement of the other.
Let XOX′ and YOY′ be a rectangular system of coordinates. Let A be any point on OX. Let ray OA be rotated in an anti clockwise direction and trace an angle θ from its initial position. Let ∠ POM = θ. Draw PM ⊥ OX. Then ΔPMO is a right angled triangle.
∠POM + ∠OPM + ∠PMO = 180°
∠POM + ∠OPM + 90° = 180°
∠POM + ∠OPM = 90°
∠OPM = 90° - ∠POM = 90° - θ
Thus, ∠OPM and ∠POM are complementary angles.
$$ \sin (90° - \theta) = \cos \theta $$
$$ \cos (90° - \theta) = \sin \theta $$
$$ \tan (90° - \theta) = \cot \theta $$
$$ \cot (90° - \theta) = \tan \theta $$
$$ \csc (90° - \theta) = \sec \theta $$
$$ \sec (90° - \theta) = \csc \theta $$
The above six results are known as trigonometric ratios of complementary angles.