Hi,

Let me take the angle Thita as 'x'. So, given is that

Cosec x + Cot x = k,

To prove that : Cos x = k^2 - 1/k^2 +1

So take the given equation we have,.

1/sinx + cosx/sinx = k,

1+cosx/sinx = k, Now we use the formula that 1+cosx = 2cos^2(x/2) and sinx = 2sin(x/2).cos(x/2)

Hence k = cot(x/2)

put value of k in the proving equation,

(k^2 -1)/ (k^2 + 1) = cot^2(x/2)  - 1 / cot^2(x/2)

this gives , = cosec^2(x -2) / cosec^2 (x/2)

when you change it in terms of sinx and use the formula of cos2x = cos^2x - sin^2x , you will get the answer cosx.

Hope it helps.