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If $\tan x / 2 = cosec x - \sin x$ , then $\sec^{2} (x / 2) =$

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a

5+1

b

5−1

c

5−2

d

5+2

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detailed solution

Correct option is B

tan⁡(x/2)=1+tan2⁡(x/2)2tan⁡(x/2)−2tan⁡(x/2)1+tan2⁡(x/2)⇒2tan2⁡(x/2)1+tan2⁡(x/2)=1+tan2⁡(x/2)2−4tan2⁡(x/2)⇒2tan4⁡(x/2)+2tan2⁡(x/2)=1+tan4⁡(x/2)−2tan2⁡(x/2)⇒tan4⁡(x/2)+4tan2⁡(x/2)−1=0⇒tan2⁡(x/2)=5−2⇒sec2⁡(x/2)=5−1

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