Question

Sec x - tan X =p then prove that cosec X =1+p^2/1-p^2

Answer 1

Sec A + Tan A =p

(SecA)^2 - (TanA)^2 = 1

(Sec A + Tan A)*(Sec A - Tan A)=1

Sec A - Tan A = 1/p

solving

Sec A = (p^2+1)/2p

Tan A = (p^2–1)/2p

Cosec A = 1/sin A = Sec A/Tan A = (p^2+1)/(p^2–1)

Answer 2

Sec x- tan x=p..(I)
Also, we know that sec^2x-tan ^2 x=1
or (sec x+tanx)(sec x -tan x)=1
(sec x+tan x)=1/p..(ii)
Adding eq. (i) and (ii) we get
2sec x=(p+1/p)
=>sec x=1/2(p+1/p)..(iii)
And tan x = -1/2(p-1/p)..(IV)
cosecx =1/sin x =(1/cos x)/(sin x/cos x)
=sec x/tan x=>1/2(p+1/p)/-1/2(p-1/p)
=>-(p^2+1)/(p^2-1)=>(1+p^2 )/(1-p^2)
Hence, Cosecx =(1+p^2)/(1-p^2)
Hope it helps