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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justaguide | College Teacher | (Level 2) Distinguished Educator

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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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