Home
Class 12
MATHS
cot^(-1)((sqrt(1+sinx)+sqrt(1-sinx))/(sq...

`cot^(-1)((sqrt(1+sinx)+sqrt(1-sinx))/(sqrt(1+sinx)-sqrt(1-sinx)))=(x)/(2), x in (0,(pi)/(4))`

Text Solution

Verified by Experts

`sqrt(1+sinx)=sqrt(sin^(2)""(x)/(2)+cos^(2)""(x)/(2)+2 sin ""(x)/(2)cos""(x)/(2))`
`=sqrt((sin""(x)/(2)+cos""(x)/(2))^(2))`
`sin""(x)/(2)+cos""(x)/(2)`
and
`sqrt(1-sinx)=sqrt(sin^(2)""(x)/(2)+cos^(2)""(x)/(2)-2 sin ""(x)/(2)cos""(x)/(2))`
`=sqrt((cos""(x)/(2)-sin""(x)/(2))^(2))=cos""(x)/(2)-sin""(x)/(2)`
`( :. x in (0,(pi)/(4))impliescos""(x)/(2) gt sin""(x)/(2))`
`=cos^(-1)""((sqrt(1+sinx)+sqrt(1-sinx))/(sqrt(1+sinx)-sqrt(1-sinx)))`
`=cot^(-1)""((sin""(x)/(2)+ cos ""(x)/(2))+(cos""(x)/(2)-sin""(x)/(2)))/((sin""(x)/(2)+ cos ""(x)/(2))-(cos""(x)/(2)-sin""(x)/(2)))`
`=cot^(-1)""((2 cos""(x)/(2))/(2 sin""(x)/(2)))`
`=cot^(-1)(cot""(x)/(2))=(x)/(2)=`RHS. Hence Proved.
Promotional Banner

Topper's Solved these Questions

  • INVERES TRIGONOMETRIC FUNCTIONS

    NAGEEN PRAKASHAN ENGLISH|Exercise Miscellaneous Exercise|8 Videos

  • INTEGRATION

    NAGEEN PRAKASHAN ENGLISH|Exercise Miscellaneous Exercise|44 Videos

  • LINEAR PROGRAMMING

    NAGEEN PRAKASHAN ENGLISH|Exercise Miscellaneous Exercise|9 Videos

Similar Questions

Explore conceptually related problems

Prove the following: cot^(-1)[(sqrt(1+sinx )+sqrt(1-sinx))/(sqrt(1+sinx)-\ sqrt(1-sinx))]=x/2,\ x (0,pi/4)

Differentiate w.r.t. x the function cot^(-1)""((sqrt(1+sinx)+sqrt(1-sinx))/(sqrt(1+sinx)-sqrt(1-sinx))),0 < x < pi/2

The value of tan^(-1)[(sqrt(1-sinx)+sqrt(1+sinx))/(sqrt(1-sinx)-sqrt(1+sinx))](AA x in [0, (pi)/(2)]) is equal to

" If "y=cot^(-1)[(sqrt(1+sinx)+sqrt(1-sinx))/(sqrt(1+sinx)-sqrt(1-sinx))](0ltxltpi//2)," then "(dy)/(dx)=

The value of cot^(-1){(sqrt(1-sinx)+sqrt(1+sinx))/(sqrt(1-sinx) -sqrt(1+sinx))} is (0 lt x lt (pi)/(2))

int(sinx)/(sqrt(1+sinx))dx

Prove that: cot^(-1){(sqrt(1+sinx)+sqrt(1-sinx))/(sqrt(1+sinx)-sqrt(1-sinx))}=pi/2-x/2 , if pi/2 < x < pi

If y="tan"^(-1)((sqrt(1+sinx)+sqrt(1-sinx)))/((sqrt(1+sinx)-sqrt(1-sinx)))," find "(dy)/(dx).

If y=cot^(-1)[(sqrt(1+sinx)+sqrt(1-sinx))/(sqrt(1+sinx)-sqrt(1-sinx))] , (0 lt x lt pi/2) , then (dy)/(dx)=

If y=cot^(-1)[(sqrt(1+sinx)+sqrt(1-sinx))/(sqrt(1+sinx)-sqrt(1-sinx))] , (0 lt x lt pi/2) , then (dy)/(dx)=