Question

tan x/2 = cosecx -sinx then cos^2 [x/2] is

Answer 1

Answer:

Tan(x/2) = cosecx - sinx

Tan(x/2) = (1/sinx) - sinx

Tan(x/2) = (1-sin²x)/sinx --------(1)

Now, we know that ,

Tan(x/2) = (1-cosx)/sinx --------(2)

Equating both we get,

(1-cosx)/sinx = (1-sin²x)/sinx = cos²x/sinx

so,

cos²x+cosx-1 = 0

solving this for cosx , we get ,

cosx = (√5+1)/4 ---------------(3)

now, we know that

cosx = 2cos²(x/2) -1

so,

2cos²(x/2) = cosx+1

cos²(x/2) = (cosx+1)/2 ----------(4)

putting value of (3) in (4th) we get,

❤️ cos²(x/2) = [(√5)+5]/8 (Ans.) ❤️

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