If A = B = 45°, show that :

sin (A - B) = sin A cos B - cos A sin B

sin (A - B) = sin A cos B - cos A sin B

L.H.S. = sin (A - B)

= sin (45° - 45°)

= sin 0°

= 0

R.H.S. = sin A cos B - cos A sin B

= sin 45° . cos 45° - cos 45°. sin 45°

$= \dfrac {1}{\sqrt2} \times \dfrac {1}{\sqrt2} - \dfrac {1}{\sqrt2} \times \dfrac {1}{\sqrt2}\\[1em] = \dfrac {1}{2} - \dfrac {1}{2}\\[1em] = 0$

∴ L.H.S. = R.H.S.

Hence, sin (A - B) = sin A cos B - cos A sin B.