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cosec x +cot x = k --------------(1)

we know that

cosec^2 x -cot^2 x = 1

(cosec x +cot x)(cosec x -cot x) = 1

k (cosec x-cot x) = 1

(cosec x-cot x) = 1/k ------------------(2)

from (1) & (2)

cosec x+cot x = k

cosec x-cot x = 1/k

2cosec x = k+1/k

2cosec x= k^2+1/k

cosec x = k^2+1/2k

sin x = 2k/k^2+1

we know that

cos x = √1-sin^2 x

cos x = √1-[2k/k^2+1]^2

cos x = √1-4k^2/(k^2+1)^2

cos x = √(k^2+1)^2-4k^2/(k^2+1)^2

cos x = √(k^2-1)^2/(k^2+1)^2 {·(a+b)^2-(a-b)^2 = 4ab}

cos x = √[k^2-1/k^2+1]^2

cos x = k^2-1/k^2+1

hence proved