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cot^(-1)(sqrtcosalpha)-tan^(-1)(sqrt(cos...

`cot^(-1)(sqrtcosalpha)-tan^(-1)(sqrt(cosalpha))`=x, x, ge 0`, then sin x equals

A

`tan^(2)((alpha)/(2))`

B

`cot^(2)((alpha)/(2))`

C

`tanalpha`

D

`cot((alpha)/(2))`

Text Solution

Verified by Experts

The correct Answer is:
A
Given , `tan^(-1)((1)/(sqrt(cosalpha)))-tan^(-1)(sqrt(cosalpha))=x`
`rArrtan^(-1)(((1)/(sqrt(cosalpha))-sqrt(cosalpha))/(1+(1)/(sqrt(cosalpha))*sqrt(cosalpha)))=x`
`rArr(1-cosalpha)/(2sqrt(cosalpha))=tanx`
`rArr(2sqrt(cosalpha))/(1-cosalpha)=cotx`
`rArrcosecx=(1+cosalpha)/(1-cosalpha)`
`rArrsinx=(1-cosalpha)/(1+cosalpha)`
`rArrsinx=(2sin^(2)((alpha)/(2)))/(2cos^(2)((alpha)/(2)))=tan^(2)((alpha)/(2))`
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