Question

if A+B+C=(2k+1)π/2, prove that cotA+cotB+cotC=cotAcotBcotC

Answer 1

$a + b + c = (2k + 1) \frac{\pi}{2}$
Hence,
$cot(a + b + c) = cot(2k + 1) \frac{\pi}{2}$
as (2k+1 )is odd so the angle will always be located in 3rd or 1 st co-ordinaye depending on the value of k. The Value of cot will always be +ve.
Now from the above equation we can derive
$(cotacotbcotc - cota - cotb - cotc) = 0$
Hence, cota+cotb+cotc=cota.cotb.cotc . (HENCE PROVED) .I have omitted some step here. but you can derive them easily.