Prove that :
LHS
= {1 + cotx – (-cosecx)} {1 + cotx + (-cosecx)}
= {1 + cotx + cosecx} {1 + cotx – cosecx}
= {(1 + cotx) + (cosecx)} {(1 + cotx) – (cosecx)}
We know that (a + b) (a – b) = a2 – b2
= (1 + cotx)2 – (cosecx)2
= 1 + cot2x + 2 cotx – cosec2x
We know that 1 + cot2x = cosec2x
= cosec2x + 2 cotx – cosec2x
= 2 cotx
= RHS
Hence proved.