The value `2tan^(-1)[sqrt((a-b)/(a+b)tantheta/2)]` is equal to `cos^(-1)((acostheta+b)/(a+bcostheta))` (b) `cos^(-1)((a+bcostheta)/(acostheta+b))` `cos^(-1)((acostheta)/(a+bcostheta))` (d) `cos^(-1)((bcostheta)/(acostheta+b))`

Options

A. `cos^(-1) ((a cos theta + b)/(a + b cos theta))`
B. `cos^(-1) ((a + b cos theta)/(a cos theta + b))`
C. `cos^(-1) ((a cos theta)/(a + b cos theta))`
D. `cos^(-1) ((b cos theta)/(a cos theta + b))`

Answer

A

Explanation

`2 tan^(-1) [sqrt((a -b)/(a + b)) "tan"(theta)/(2)]`
`= cos^(-1) [(1 - ((a -b)/(a + b)) tan^(2).(theta)/(2))/(1 + ((a -b)/(a +b))tan^(2).(theta)/(2))] " " [ :' 2 tan^(-1) x = cos^(-1).(1 -x^(2))/(1 + x^(2))]`
`= cos^(-1) [((a + b) -(a -b) tan^(2).(theta)/(2))/((a + b) + (a -b) tan^(2).(theta)/(2))]`
`= cos^(-1) [(a (1 - tan^(2).(theta)/(2)) + b(1 + tan^(2).(theta)/(2)))/(a(1 + tan^(2).(theta)/(2)) + b (1 - tan^(2).(theta)/(2)))]`
`= cos^(-1) [((a( 1 - tan^(2).(theta)/(2)))/(1 + tan^(2).(theta)/(2)) + b)/(a + b ((1 - tan^(2).(theta)/(2))/(1 + tan^(2).(theta)/(2))))]`
`= cos^(-1) [(a cos theta + b)/(a + b cos theta)]`