LHS = tan (-225°) cot (-405°) – tan (-765°) cot (675°)
We know that tan (-x) = -tan (x) and cot (-x) = -cot (x).
= [-tan (225°)] [-cot (405°)] – [-tan (765°)] cot (675°)
= tan (225°) cot (405°) + tan (765°) cot (675°)
= tan (90° × 2 + 45°) cot (90° × 4 + 45°) + tan (90° × 8 + 45°) cot (90° × 7 + 45°)
= tan 45° cot 45° + tan 45° [-tan 45°]
= 1 × 1 + 1 × (-1)
= 1 – 1
= 0
= RHS
Hence proved.