2008/003) Prove that the sum of

1/(sinA + sin2A) + 1/(sin2A + sin3A) + 1/(sin3A + sin4A)........to n terms

IS

cosecA[cotA - cot(n+1)A]

proof:
We know
1/( sin ka sin (k+1) a)
= cosec a sin a /( sin ka sin (k+1) a)
= cosec a ( sin ((k+1) a - ka))/( sin ka sin (k+1) a)
= cosec a ( sin (k+1) a cos ka - cos(k+1)a sin ka) / sin (k+1) a sin ka
= cosec a ( cos ka/ sin ka - cos (k+1)a / sin (k+1) a)
= cosec a ( cot ka - cot (k+1) a)

addding from k = 1 to n we get the result.