The value of `tan^(-1)((xcostheta)/(1-xsintheta))-cot^(-1)((costheta)/(x-sintheta))i s` `2theta` (b) `theta` (c) `theta/2` (d) independent of `theta`

Options

A. `2 theta`
B. `theta`
C. `theta//2`
D. independent of `theta`

Answer

B

Explanation

`tan^(-1) ((x cos theta)/(1 - x sin theta)) - cot^(-1) ((cos theta)/(x - sin theta))`
`= tan^(-1) ((x cos theta)/(1 - x sin theta)) - tan^(-1) ((x - sin theta)/(cos theta))`
`= tan^(-1) (((x cos theta)/(1-x sin theta) -(x - sin theta)/(cos theta))/(1 + ((x cos theta)/(1 -x sin theta)) ((x-sin theta)/(cos theta))))`
`= tan^(-1) ((x cos^(2) theta - x + sin theta + x^(2) sin theta - x sin^(2) theta)/(cos theta - x cos theta sin theta + x^(2) cos theta - x cos theta sin theta))`
`= tan^(-1) ((-x sin^(2) theta + sin theta + x^(2) sin theta - x sin^(2) theta)/(cos theta - 2x cos theta sin theta + x^(2) cos theta))`
`= tan^(-1) ((-2 x sin^(2) theta + sin theta + x^(2) sin theta)/(cos theta -2 x cos theta sin theta+ x^(2) cos theta))`
`= tan^(-1) ((sin theta(-2 x sin theta + 1 + x^(2)))/(cos theta (1-2x sin theta + x^(2))))`
`= tan^(-1) (tan theta)`
`= theta`