Question

cosec thita +cot thita=p prove that costheta=p squar-1/p squar +1

Answer 1

(Cosecx+cotx) (Cosecx-cotx)=1 Cosecx-cotx=1/p Adding (Cosecx+cotx)and (Cosecx-cotx) 2cosecx=p + 1/p Cosecx=(p^2+1)/2p

Answer 2

p=cosecΘ+cotΘ
On squaring,
p^2=(cosecΘ+cotΘ)(cosecΘ+cotΘ)
      =cosec^2Θ+cot^2Θ+2cosecΘ·cotΘ
p^2- 1 =cosΘ
p^2+1
LHS
p^2- 1 
p^2+1
On substituting
cosec^2Θ+cot^2Θ+2cosecΘ·cotΘ-1
cosec^2Θ+cot^2Θ+2cosecΘ·cotΘ+1
(cosec^2Θ-1)+cot^2Θ+2cosecΘ·cotΘ
cosec^2Θ+(cot^2Θ+1)+2cosecΘ·cotΘ
cot^2Θ+cot^2Θ+2cosecΘ·cotΘ                          {Because 1+cot^2Θ=cosec^2Θ}
cosec^2Θ+cosec^2Θ+2cosecΘ·cotΘ
2cot^2Θ+2cosecΘ·cotΘ
2cosec^2Θ+2cosecΘ·cotΘ
2cotΘ(cotΘ+cosecΘ)
2cosecΘ(cosecΘ+cotΘ)
cotΘ
cosecΘ
cosΘ/sinΘ
1/sinΘ
cosΘ  ·  sinΘ
sinΘ         1
cosΘ
Therefore LHS=RHS.
So p^2- 1 =cosΘ
      p^2+1