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Question
Select the correct option from the given alternatives:
The value of `costheta/(1 + sin theta)` is equal to .....
Options
`tan(theta/2 - pi/4)`
`tan(-pi/4 - theta/2)`
`tan(pi/4 - theta/2)`
`tan(pi/4 + theta/2)`
Solution
`tan(pi/4 - theta/2)`
Explanation:
`costheta/(1 + sintheta) = (cos^2 theta/2 - sin^2 theta/2)/(cos^2 theta/2 + sin^2 theta/2 + 2sin theta/2 cos theta/2)`
= `((cos theta/2 - sin theta/2)(cos theta/2 + sin theta/2))/((cos theta/2 + sin theta/2)^2`
= `(cos theta/2 - sin theta/2)/(cos theta/2 + sin theta/2)`
Dividing numerator and denominator by `cos theta/2`, we get
`costheta/(1 + sintheta) = (1 - tan theta/2)/(1 + tan theta/2) = tan(pi/4 - theta/2)`
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