If `tan^(-1){(sqrt(1+x^2)-sqrt(1-x^2))/(sqrt(1+x^2)+sqrt(1-x^2))}=alpha` , then `x^2=` `sin2\ alpha` (b) `sin\ alpha` (c) `cos2\ alpha` (d) `cos\ alpha`

Explanation

`tan^(-1){(sqrt(1+x^2)-sqrt(1-x^2))/(sqrt(1+x^2)+sqrt(1-x^2))}=alpha`

`tan alpha={(sqrt(1+x^2)-sqrt(1-x^2))/(sqrt(1+x^2)+sqrt(1-x^2))}` and `x=sqrt(sin 2t)`

`tan alpha={(sqrt(sin^2t+cos^2t+sin2t)-sqrt(sin^2t+cos^2t-sin2t))/(sqrt(sin^2t+cos^2t+sin2t)+sqrt(sin^2t+cos^2t-sin2t))}`

`tan alpha={(sqrt((cost+sint)^2)-sqrt((cost-sint)^2))/((sqrt((cost+sint)^2))+sqrt((cost-sint)^2))))}`

`=frac{cost+sint-cost+sint}{cost+sint+cost-sint}`

`=frac{2 sint}{2 cost}`

`=tan t`

`alpha=t`
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