Prove that : (cot(x))/(1-sin^2(x)) + (cos(x))/ (csc^2(x)-1) = (sec (x)) (sin^2(x)+ csc (x))

We have to prove (cot(x))/(1-sin^2(x)) + (cos(x))/
(csc^2(x)-1) = (sec (x)) (sin^2(x)+ csc
(x))

(cot(x))/(1-sin^2(x)) + (cos(x))/
(csc^2(x)-1)

=> (cot x/(cos x)^2) + (cos(x))/ ((1/
(sin x)^2) -1)

=> (cot x/(cos x)^2) + (cos(x))/ ((1
- (sin x)^2)/ (sin x)^2)

=> (cot x/(cos x)^2) +
(cos(x))/ [(cos x)^2 / (sin x)^2)]

=> (cot x/(cos
x)^2) + (cos(x) * (sin x)^2/ (cos x)^2

=> (cos x/
sin x)/ (cos x)^2) + (cos(x) * (sin x)^2)/ (cos
x)^2

=> 1 / (sin x*cos x) + (sin x)^2/ cos
x

=> 1 / (sin x*cos x) + (sin x)^3/ cos x * sin
x

=> [1 + (sin x)^3] / (cos x * sin x)
...(1)

(sec (x)) (sin^2(x)+ csc
(x))

=> (1/ cos x)*[( sin x)^2) + 1/sin
x]

=> (1/ cos x)*[(sin x)^3 + 1]/ sin
x

=> [1 + (sin x)^3]/( sin x* cos x)
...(2)

From (1) and (2), we see that they are
equal.

This proves that (cot(x))/(1-sin^2(x))
+ (cos(x))/ (csc^2(x)-1) = (sec (x)) (sin^2(x)+ csc
(x))