If cosec θ + cot θ = p, then \(\rm\frac{p^2-1}{p^2+1}\) equal:

Formula Used:

(a + b)² = a² + 2ab + b²,

sin2 θ+cos² θ = 1 and 1 - sin2 θ = cos² θ

cot θ = cos θ/sin θ and cosec θ = 1/sin θ

and for equations a/b = c/d

Using the Componendo and Dividendo rule, we get 

(a + b)/(a - b) = (c + d)/(c - d)

or (a-b)/(a+b) = (c-d)/(c+d)

Calculation:

1/sin θ + cos θ/sin θ = p

⇒ (1+cos θ)/sin θ = p/1

Squaring both sides of the given equation:
(1 + cos θ)²/sin² θ = p²/1

⇒ (1 + cos² θ + 2cos θ)/sin2 θ = p²/1

Applying Componendo and Dividendo, we get

(1 + cos² θ + 2cos θ - sin2 θ)/(1 + cos² θ + 2cos θ + sin2 θ) = (p²-1)/(p²+1)

⇒ (1 - sin2 θ + cos² θ + 2cos θ )/(1 + cos² θ+ sin2 θ + 2cos θ ) = (p²-1)/(p2+1)

⇒ (cos² θ + cos² θ + 2cos θ )/(1 + 1 + 2cos θ ) = (p²-1)/(p2+1)

⇒ (2cos² θ + 2cos θ )/(2 + 2cos θ ) = (p²-1)/(p2+1)

⇒ 2cos θ(cos θ + 1 )/2(1 + cos θ ) = (p²-1)/(p2+1)

⇒ cos θ = (p²-1)/(p2+1)

Thus, value of  (p²-1)/(p2+1) is cos θ.