निम्नलिखित में से प्रत्येक का मान ज्ञात कीजिए :
` tan""(1)/(2)[sin^(-1)""(2x)/(1+x^(2))+cos^(-1)""(1-y^(2))/(1+y^(2))],|x| lt 1 , y gt 0` तथा `xy lt 1`

Options

A. `(x+y)/(1-xy)`
B. `(x-y)/(1+xy)`
C. `(xy)/(x+y)`
D. `(xy)/(x-y)`

Answer

A

Explanation

We have , `"tan"(1)/(2)("sin"^(-1)(2x)/(1+x^(2))+"cos"^(-1)(1-y^(2))/(1+y^(2)))`
`="tan"(1)/(2)(2tan^(-1)x+2tan^(-1)y)`
`[because2tan^(-1)x=sin^(-1)((2x)/(1+x^(2)))=cos^(-1)((1-x^(2))/(1+x^(2)))]`
`="tan"(1)/(2)xx2(tan^(-1)x+tan^(-1)y)`
`="tan"["tan"^(-1)((x+y)/(1-xy))]`
`=(x+y)/(1-xy)[{:(becausetan^(-1)x+tan^(-1)y=tan^(-1)y,),(="tan"^(-1)((x+y)/(1-xy)),):}]`