If $cosec~ \theta+\cot \theta=p$, then the value of $\frac{p^2-1}{p^2+1}$ is:

$\cos \theta$

We are given that :-

cosec θ + cot θ = p    -----(1)

{ we know, cosec² θ - cot² θ  = 1

So, cosec θ - cot θ = \(\frac{1 }{cosec θ + cot θ}\) }

So, cosec θ - cot θ = \(\frac{1 }{p}\)        ----(2)

On adding equation 1 and 2 .

2 cosec θ = p + \(\frac{1 }{p}\)

2 cosec θ =  \(\frac{ P² + 1 }{p}\)    ------(3)

On subtracting equation 1 from 2 .

2 cot θ = p - \(\frac{1 }{p}\)

2 cot θ =  \(\frac{ P² -1 }{p}\)      ----(4)

Subtracting euation 4 by equation 3

cos θ = \(\frac{ P² -1 }{p}\) × \(\frac{ P }{P² +1}\) 

cos θ = \(\frac{ P² -1 }{p²+1}\)

So, Ans :- cosθ