arcsin-x-arcsin-1-x-t-sin-x-sin-1-x-t-sin-cos-cos-sin-sin-t-x-1-1-x-1-x-1-x-x-1-x-1-t-pi-2-

Question Number 136679 by mnjuly1970 last updated on 24/Mar/21

a r c s i n \sqrt{x} + a r c s i n (\sqrt{1 - x}) = t

s i n (α) = \sqrt{x} s i n (β) = \sqrt{1 - x}

α + β = t

s i n (α) c o s (β) + c o s (α) s i n (β) = s i n (t)

\sqrt{x} . \sqrt{1 - 1 + x} + \sqrt{1 - x} . \sqrt{1 - x}

= x + 1 - x = 1

t = \frac{π}{2} \dots

Answered by Olaf last updated on 25/Mar/21

Let f (x) = \arcsin \sqrt{x} + \arcsin \sqrt{1 - x}

f^{'} (x) = \frac{1}{2 \sqrt{x}} . \frac{1}{\sqrt{1 - x}} - \frac{1}{2 \sqrt{1 - x}} . \frac{1}{\sqrt{1 - (1 - x)}}

f^{'} (x) = \frac{1}{2 \sqrt{x}} . \frac{1}{\sqrt{1 - x}} - \frac{1}{2 \sqrt{1 - x}} . \frac{1}{\sqrt{x}}

f^{'} (x) = 0

\Rightarrow f (x) = cste = f (0) = \arcsin (0) + \arcsin (1)

= 0 + \frac{π}{2} = \frac{π}{2}

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