If cosec theta + cot theta = K, then prove that cos theta = K^2 -1/K^2 +1
Answer (1)
3rd Jan, 2021
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.
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