Proof-cot-1-1-sint-1-sint-1-sint-1-sint-t-2-

Question Number 193467 by aba last updated on 14/Jun/23

Proof :

\cot^{- 1} (\frac{\sqrt{1 + sint} + \sqrt{1 - sint}}{\sqrt{1 + sint} - \sqrt{1 - sint}}) = \frac{t}{2}

Commented by MM42 last updated on 14/Jun/23

a t t e n t i o n

i n a l l t h r e e a r g u m e n t s w e u s e d e q u a l i t i e s

t h a t t h e r e l a t i o n s h i p s u c h a s

“ \sqrt{1 - {s i n}^{2} x} = c o s x^{″} o r “ {(s i n x + c o s x)}^{2} = 1 + s i n 2 x^{″} o r “ {(s i n x - c o s x)}^{2} = 1 - s i n 2 x^{″}

w h i c h m a y n o t a l w a y s h o l d .

Commented by Frix last updated on 15/Jun/23

Yes . We get

\cot^{- 1} \frac{1 + ∣ \cos t ∣}{\sin t} = \frac{t}{2}

Which is only true for

- \frac{π}{2} ⩽ t ⩽ \frac{π}{2}

Commented by MM42 last updated on 15/Jun/23

a t t e n t i o n

i f f (x) = t a n x \Rightarrow f o r \forall x \in (\frac{(2 k - 1) π}{2}, \frac{(2 k + 1) π}{2}) \to R_{f} = R

\Rightarrow \exists f^{- 1} (x) = {t a n}^{- 1} (x) & D_{f^{- 1}} = R & R_{f^{- 1}} = (- \frac{π}{2}, \frac{π}{2})

t a n : (\frac{(2 k - 1) π}{2}, \frac{(2 k + 1) π}{2}) \to R \Leftrightarrow {t a n}^{- 1} : R \to (- \frac{π}{2}, \frac{π}{2})

i f f (x) = c o t x \Rightarrow f o r \forall x \in (k π, (k + 1) π) \to R_{f} = R

\Rightarrow \exists f^{- 1} (x) = {c o t}^{- 1} (x) & D_{f^{- 1}} = R & R_{f^{- 1}} = (0, π)

c o t : (k π, (k - 1) π) \to R \Leftrightarrow {c o t}^{- 1} : R \to (0, π)

t h e r e f o r e i f {c o t}^{- 1} (\frac{}{}) = \frac{t}{2} \Rightarrow

0 < \frac{t}{2} < π \Rightarrow 0 < t < 2 π

Answered by Frix last updated on 14/Jun/23

\frac{\sqrt{1 + s} + \sqrt{1 - s}}{\sqrt{1 + s} - \sqrt{1 - s}} =

= \frac{{(\sqrt{1 + s} + \sqrt{1 - s})}^{2}}{(\sqrt{1 + s} - \sqrt{1 - s}) (\sqrt{1 + s} + \sqrt{1 - s})} =

= \frac{1 + \cos t}{\sin t} = \cot \frac{t}{2}

Answered by Subhi last updated on 14/Jun/23

(\frac{\sqrt{1 + s i n (t)} + \sqrt{1 - s i n (t)}}{\sqrt{1 + s i n (t)} - \sqrt{1 - s i n (t)}}) . (\frac{\sqrt{1 + s i n (t)}}{\sqrt{1 + s i n (t)}}) = c o t (\frac{t}{2})

\frac{1 + s i n (t) + \sqrt{1 - {s i n}^{2} (t)}}{1 + s i n (t) - \sqrt{1 - {s i n}^{2} (t)}} = \frac{1 + s i n (t) + c o s (t)}{1 + s i n (t) - c o s (t)}

\frac{1 + 2 s i n (\frac{t}{2}) . c o s (\frac{t}{2}) + 2 {c o s}^{2} (\frac{t}{2}) - 1}{1 + 2 s i n (\frac{t}{2}) c o s (\frac{t}{2}) - 1 + 2 {s i n}^{2} (\frac{t}{2})}

\frac{2 c o s (\frac{t}{2}) (s i n (\frac{t}{2}) + c o s (\frac{t}{2}))}{2 s i n (\frac{t}{2}) (s i n (\frac{t}{2}) + c o s (\frac{t}{2}))} = c o t (\frac{t}{2})

Answered by MM42 last updated on 14/Jun/23

S 1

A = {c o t}^{- 1} (\frac{\sqrt{1 + s i n t} + \sqrt{1 - s i n t}}{\sqrt{1 + s i n t} - \sqrt{1 - s i n t}}) =

{c o t}^{- 1} (\frac{\sqrt{{(c o s \frac{t}{2} + s i n \frac{t}{2})}^{2}} + \sqrt{{(c o s \frac{t}{2} - s i n \frac{t}{2})}^{2}}}{\sqrt{{(c o s \frac{t}{2} + s i n \frac{t}{2})}^{2}} - \sqrt{{(c o s \frac{t}{2} - s i n \frac{t}{2})}^{2}}})

\Rightarrow A = {c o t}^{- 1} (\frac{c o s \frac{t}{2}}{s i n \frac{t}{2}}) = \frac{t}{2}

S 2 l e t t a n \frac{t}{2} = u

A = {c o t}^{- 1} (\frac{\sqrt{1 + \frac{2 u}{1 + u^{2}}} + \sqrt{1 - \frac{2 u}{1 + u^{2}}}}{\sqrt{1 + \frac{2 u}{1 + u^{2}}} - \sqrt{1 - \frac{2 u}{1 + u^{2}}}}) =

{c o t}^{- 1} (\frac{\sqrt{{(1 + u)}^{2}} + \sqrt{{(1 - u)}^{2}}}{\sqrt{{(1 + u)}^{2}} - \sqrt{{(1 - u)}^{2}}}) =

{c o t}^{- 1} (\frac{2}{2 u}) = {c o t}^{- 1} (u) = \frac{t}{2}

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