Home
Class 12
MATHS
Prove that:tan^(-1)sqrt(x)=1/2cos^(-1)((...

Prove that:`tan^(-1)sqrt(x)=1/2cos^(-1)((1-x)/(1+x)), x in [0,1]`

Text Solution

Verified by Experts

Let `sqrt(x)=tan thetaimpliestan^(2)theta`
`:. RHS=(1)/(2) cos^(-1)""((1-x)/(1+x)), " " x in [0,1]`
`=(1)/(2) cos^(-1)""((1-tan^(2) theta)/(1+tan^(2)theta))`
`=(1)/(2)cos^(-1)(cos2theta)=(1)/(2)(2theta)`
`=theta=tan^(-1)sqrt(x)=` LHS Hence Proved.
Doubtnut Promotions Banner Mobile Dark
|

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: tan^(-1)sqrt(x)=1/2cos^(-1)((1-x)/(1+x)),\ x\ \ (0,1)

Prove that: sin{tan^(-1)((1-x^2)/(2x))+cos^(-1)((1-x^2)/(1+x^2))}=1

Prove that: sin{tan^(-1)((1-x^2)/(2x))+cos^(-1)((1-x^2)/(1+x^2))}=1

Prove that tan^(-1){x/(a+sqrt(a^2-x^2))}=1/2sin^(-1)x/a ,-a lt x lt a

Prove that: tan^(-1)[(sqrt(1+x)-sqrt(1-x))/(sqrt(1+x)+sqrt(1-x))]=pi/4-1/2cos^(-1)x ,\ -1/(sqrt(2))\ lt=x\ lt=1

Prove that: tan^(-1)((1-x)/(1+x))-tan^(-1)((1-y)/(1+y))=sin^(-1)((y-x)/(sqrt((1+x^2) (1+y^2))))

Prove that: tan^(-1)((1-x)/(1+x))-tan^(-1)((1-y)/(1+y))=sin^(-1)((y-x)/(sqrt((1+x^2)(1+y^2))))

Prove that 1/2 cos^(-1) ((1-x)/(1+x))=tan^(-1) sqrtx .

Prove that: tan^(-1){(sqrt(1+x)-sqrt(1-x))/(sqrt(1+x)+sqrt(1-x))}=pi/4-1/2. cos^(-1)x , 0

Prove that: tan^(-1) {(sqrt(1+x)-sqrt(1-x))/(sqrt(1+x)+sqrt(1-x))} = pi/4-1/2\ cos^(-1)x

Home
Profile