Home
Class 12
MATHS
Prove thattan^(-1)((sqrt(1+x)-sqrt(1-x))...

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=xlt=1`

Text Solution

Verified by Experts

`L.H.S. = tan^-1[(sqrt(1+x)-sqrt(1-x))/(sqrt(1+x)+sqrt(1-x))]`
`=tan^-1[(sqrt(1+x)(1-sqrt(1-x)/sqrt(1+x)))/(sqrt(1+x)(1+sqrt(1-x)/sqrt(1+x)))]`
`=tan^-1[(1-sqrt(1-x)/sqrt(1+x))/(1+sqrt(1-x)/sqrt(1+x))]`
As, `tan^-1x-tan^-1y = (x-y)/(1+xy)`
So, our expression becomes,
`=tan^-1(1)+tan^-1(sqrt(1-x)/sqrt(1+x))`
`=pi/4+1/2(2tan^-1(sqrt(1-x)/sqrt(1+x)))`
Also, `2tan^-1y = cos^-1((1-y^2)/(1+y^2))`
...
Promotional Banner

Similar Questions

Explore conceptually related problems

Prove that tan^(-1)((sqrt(1+x)-sqrt(1-x))/(sqrt(1+x)-sqrt(1-x)))=pi/4-1/2cos^(-1),-1/(sqrt(2))lt=xlt=1

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

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^2)+sqrt(1-x^2))/(sqrt(1+x^2)-sqrt(1-x^2))]=pi/4+1/2cos^(-1)x^2

y=tan^(-1)((sqrt(1+x^2)+sqrt(1-x^2))/(sqrt(1+x^2)-sqrt(1-x^2)))

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

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

tan^(-1)((sqrt(1+x^(2))+sqrt(1-x^(2)))/(sqrt(1+x^(2))-sqrt(1-x^(2))))

If y=tan^(-1)((sqrt(1+x^2)-sqrt(1-x^2))/(sqrt(1+x^2)+sqrt(1-x^2))) find (dy)/(dx)