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