MHB Simplifying Expression: Simplify Expression

[anemone]

Simplify the expression $\dfrac{\sqrt{1+\sqrt{1-a^2}}((1+a)\sqrt{1+a}-(1-a)\sqrt{1-a})}{a(2+\sqrt{1-a^2})}$.

 

[Albert1]

$\sqrt 2$
Is it correct ?

 

[anemone]

$\sqrt 2$
Is it correct ?

Yes, $\sqrt{2}$ is the correct answer. :)

 

[Albert1]

Simplify the expression $\dfrac{\sqrt{1+\sqrt{1-a^2}}((1+a)\sqrt{1+a}-(1-a)\sqrt{1-a})}{a(2+\sqrt{1-a^2})}---(1)$.

Spoiler

let :$x=\sqrt{1+a}, y=\sqrt{1-a}$
(1)becomes :$\dfrac{2\sqrt{1+xy}(x^3-y^3)}{(x^2-y^2)(2+xy)}$
$=\dfrac {2\sqrt{1+xy}}{x+y}=\dfrac{2}{\sqrt 2}=\sqrt 2$
for :$x^2+y^2=2, x^2-y^2=2a, (x+y)=\sqrt {2(1+xy)}$