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The value of tan^(-1)((xcostheta)/(1-xsi...

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`

A

`2 theta`

B

`theta`

C

`theta//2`

D

independent of `theta`

Text Solution

Verified by Experts

The correct Answer is:
B
`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`
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CENGAGE-INVERSE TRIGONOMETRIC FUNCTIONS-Exercise (Single)
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  2. If cot^(-1)x+cot^(-1)y+cot^(-1)z=pi/2,x , y , z >0a n dx y<1, then x+y...

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  3. If x^2+y^2+z^2=r^2,t h e ntan^(-1)((x y)/(z r))+tan^(-1)((y z)/(x r))+...

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  4. The value of tan^(-1)((xcostheta)/(1-xsintheta))-cot^(-1)((costheta)/(...

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  5. if cot^(-1)[sqrt(cosalpha)]-tan^(-1)[sqrt(cosalpha)]=x then sinx is eq...

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  9. The sum of series sec^(-1)sqrt(2)+sec^(-1)(sqrt(10))/3+sec^(-1)(sqrt(5...

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  10. If (1)/(2) sin^(-1) [(3 sin 2 theta)/(5 + 4 cos 2 theta)] = tan^(-1) x...

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  11. The value 2tan^(-1)[sqrt((a-b)/(a+b)tantheta/2)] is equal to cos^(-1)(...

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  18. If sin^(-1)x+sin^(-1)y+sin^(-1)z=pi,t h e nx^4+y^2+z^4+4x^2y^2z^2=K(x^...

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