Cosec x =1/sin x

Cot x = cos x/sin x

Cosec x ? cot x = (1-cos x)/sin x

=>1-cos x = p sin x

P+1/p = (1-cos x)/ sin x + sin x /1-cos x

_____= [1+cos² x ? 2 cos x + sin² x] / (sin x) (1-cos x) ------------by taking LCM

_____= [1+1-2 cos x] / (sin x) (1-cos x) ----------since sin² x+ cos² x =1

_____= [2-2 cos x] / (sin x) (1-cos x) -------by taking ?2? common

_____= 2{1-cos x}/ (sin x) (1-cos x) -------- (1-cos x) in numerator & denominator

_____= 2 / sin x

_____= 2 cosec x

Therefore, cosec x = ½ (p + 1/p)