Here we have to prove sinQ/cotQ + cosecQ = 2 + sinQ/cotQ – cosecQ we will start with LHS (Left hand side ie sinQ/cotQ + cosecQ ) and prove that it is equal to 2 + sinQ/cotQ – cosecQ Hence we will get LHS = RHS
Our aim is to Show that sinQ/cotQ + cosecQ = 2 + sinQ/cotQ – cosecQ
Proof
Lets start with LHS of the equations
LHS = sinQ/(cotQ + cosecQ)
= sinQ/(cosQ/sinQ + 1/sinQ)
= sinQ2 /(cosq+1)/sinQ
= sinQ2 /(1 + cosQ)
= sin²Q (1-cosQ) /(1 + cosQ)(1 – cosQ)
= sin²Q (1-cosQ) /(1 – cos2Q)
= sin²Q (1-cosQ) /sin²Q Hint : (1 – cos²Q) = sin²Q since sin²Q + cos²Q = 1
= 1-cosQ
RHS = 2 + sinQ/cotQ – cosecQ
= 2+ sinQ/(cosQ/sinQ-1/sinq)
= 2 +sin²q/(cosQ – 1)
= 2 -sin²q/(1-cosQ )
= 2 – sin²q(1 + cosQ)/(1-cosQ) (1 + cosQ)
= 2 – sin²q(1 + cosQ)/(1-cos²Q)
= 2 – sin²q(1 + cosQ)/sin²q Hint : (1 – cos²Q) = sin²Q since sin²Q + cos²Q = 1
= 2 – (1 + cosQ)
= 1 – cosQ
RHS = LHS
Hence proved
