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Answer 1

$\huge \sf{ \underline{ \underline{Question : - }}}$

Prove That:-

$\large \sf \cot^{ - 1} ( \frac{ \sqrt{1 + \sin \: x } + \sqrt{1 - \sin x} }{ \sqrt{1 - \sin \: x } - \sqrt{1 - \sin \: x } } = \frac{x}{2} ;∈(0, \frac{\pi}{4} )$

$\huge \sf{ \underline{ \color{maroon}{Solution:- }}}$

$\large \sf \: L.H.S = \ \cot^{ - 1} ( \frac{ \sqrt{1 + \sin \: x } + \sqrt{1 - \sin x} }{ \sqrt{1 - \sin \: x } - \sqrt{1 - \sin \: x } }$

$\sf \: ⇒ \cot ^{ - 1} ( \frac{ \sqrt{1 + \sin \:x } \: + \sqrt{1 - \sin \: x} }{ \sqrt{1 + \sin \: x } - \sqrt{1 - \sin \: x } } \times \frac{( \sqrt{1 + \sin \: x } + \sqrt{1 + \sin \: x }) }{( \sqrt{1 + \sin \: x } + \sqrt{1 - \sin \: x } }$

$⇒ \cot ^{ - 1} ( \frac{1 + \sin \: x + 1 - \sin \: x + 2 \sqrt{1 - \sin \: x } }{1 + \sin \: x - 1 + \sin \: x } )$

$⇒ \cot ^{ - 1} ( \frac{2 + 2 \cos \: x}{2 \sin \: x } )$

$\boxed{ \color{maroon}{⇒ \cot^{ - 1} ( \frac{1 + \cos \: x }{ \sin \: x } )}}$

$⇒ \bf\cot ^{ - 1} ( \frac{2 \cos \: x \frac{x}{2} }{2 \sin \frac{x}{2} \cos \frac{x}{2} } )$

$⇒ \cot ^{ - 1}( { \cot}{ \frac{x}{2} } )$

$\boxed{ \color{goldenrod}{⇒ \frac{x}{2} = \bf \: R.H.S}}$

$\huge \pink \dag \: { \large{ \boxed{ \red{Hence \: proved}}}}$

Answer 2

Answer:

Prove That:-

\large \sf \cot^{ - 1} ( \frac{ \sqrt{1 + \sin \: x } + \sqrt{1 - \sin x} }{ \sqrt{1 - \sin \: x } - \sqrt{1 - \sin \: x } } = \frac{x}{2} ;∈(0, \frac{\pi}{4} )cot

−1

(

1−sinx

−

1−sinx

1+sinx

+

1−sinx

=

2

x

;∈(0,

4

π

)

\huge \sf{ \underline{ \color{maroon}{Solution:- }}}

Solution:−

\large \sf \: L.H.S = \ \cot^{ - 1} ( \frac{ \sqrt{1 + \sin \: x } + \sqrt{1 - \sin x} }{ \sqrt{1 - \sin \: x } - \sqrt{1 - \sin \: x } }L.H.S= cot

−1

(

1−sinx

−

1−sinx

1+sinx

+

1−sinx

\sf \: ⇒ \cot ^{ - 1} ( \frac{ \sqrt{1 + \sin \:x } \: + \sqrt{1 - \sin \: x} }{ \sqrt{1 + \sin \: x } - \sqrt{1 - \sin \: x } } \times \frac{( \sqrt{1 + \sin \: x } + \sqrt{1 + \sin \: x }) }{( \sqrt{1 + \sin \: x } + \sqrt{1 - \sin \: x } }⇒cot

−1

(

1+sinx

−

1−sinx

1+sinx

+

1−sinx

×

(

1+sinx

+

1−sinx

(

1+sinx

+

1+sinx

)

⇒ \cot ^{ - 1} ( \frac{1 + \sin \: x + 1 - \sin \: x + 2 \sqrt{1 - \sin \: x } }{1 + \sin \: x - 1 + \sin \: x } )⇒cot

−1

(

1+sinx−1+sinx

1+sinx+1−sinx+2

1−sinx

)

⇒ \cot ^{ - 1} ( \frac{2 + 2 \cos \: x}{2 \sin \: x } )⇒cot

−1

(

2sinx

2+2cosx

)

\boxed{ \color{maroon}{⇒ \cot^{ - 1} ( \frac{1 + \cos \: x }{ \sin \: x } )}}

⇒cot

−1

(

sinx

1+cosx

)

⇒ \bf\cot ^{ - 1} ( \frac{2 \cos \: x \frac{x}{2} }{2 \sin \frac{x}{2} \cos \frac{x}{2} } )⇒cot

−1

(

2sin

2

x

cos

2

x

2cosx

2

x

)

⇒ \cot ^{ - 1}( { \cot}{ \frac{x}{2} } )⇒cot

−1

(cot

2

x

)

\boxed{ \color{goldenrod}{⇒ \frac{x}{2} = \bf \: R.H.S}}

⇒

2

x

=R.H.S

\huge \pink \dag \: { \large{ \boxed{ \red{Hence \: proved}}}}†

Henceproved

Step-by-step explanation:

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