Prove that -1-1- sin x - -1- sin x-1- sin x- -1- sin x - x2- 0-x-2-

LHS =^ 1(√(1+sin x)+√(1 sin x)√(1+sin x) √(1 sin x)) sin x=2sinx2cosx2 ⇒^ 1(√((cosx2+sinx2)^2)+√((cosx2 sinx2)^2)√((cosx2+sinx2)^2) √((cosx2 sinx2 ...

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Standard XII

Mathematics

Domain and Range of Basic Inverse Trigonometric Functions

Prove that ...

Question

Prove that ${cot}^{- 1} {\frac{\sqrt{1 + sin x} + \sqrt{1 - sin x}}{\sqrt{1 + sin x} - \sqrt{1 - sin x}}} = \frac{x}{2}, 0 < x < \frac{π}{2} .$

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Solution

LHS $= {cot}^{- 1} (\frac{\sqrt{1 + sin x} + \sqrt{1 - sin x}}{\sqrt{1 + sin x} - \sqrt{1 - sin x}})$
$sin x = 2 sin \frac{x}{2} cos \frac{x}{2}$
$\Rightarrow {cot}^{- 1} (\frac{\sqrt{(cos \frac{x}{2} + sin \frac{x}{2})^{2}} + \sqrt{(cos \frac{x}{2} - sin \frac{x}{2})^{2}}}{\sqrt{(cos \frac{x}{2} + sin \frac{x}{2})^{2}} - \sqrt{(cos \frac{x}{2} - sin \frac{x}{2})^{2}}}$
$\Rightarrow {cot}^{- 1} (\frac{cos \frac{x}{2} + sin \frac{x}{2} + cos \frac{x}{2} - sin \frac{x}{2}}{cos \frac{x}{2} + sin \frac{x}{2} - cos \frac{x}{2} + sin \frac{x}{2}})$
$\Rightarrow {cot}^{- 1} (\frac{2 cos \frac{x}{2}}{2 sin \frac{x}{2}})$
$\Rightarrow {cot}^{- 1} (cot \frac{x}{2})$
$= \frac{x}{2} = R H S$

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Similar questions

{cot}^{- 1} (\frac{\sqrt{1 + sin x} + \sqrt{1 - sin x}}{\sqrt{1 + sin x} - \sqrt{1 - sin x}}) = \frac{x}{2}, x \in (0, \frac{π}{4})

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

=....

(0 < x < \frac{π}{2})

Q. If

{cot}^{- 1} (\frac{\sqrt{1 + sin x} + \sqrt{1 - sin x}}{\sqrt{1 + sin x} - \sqrt{1 - sin x}}) = \frac{x}{m}, x \in [0, \frac{π}{2}]

. Find

m

Q. If

y = {cot}^{- 1} \frac{\sqrt{1 + sin x} + \sqrt{1 - sin x}}{\sqrt{1 + sin x} - \sqrt{1 - sin x}}

, find

\frac{d y}{d x}

x \in (0, \frac{π}{2}) \cup (\frac{π}{2}, π)

S h o w t h a t

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

f o r x \in (0, \frac{π}{2})

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Prove that cot−1{√1+ sinx+√1− sinx√1+ sinx−√1− sinx}=x2,0<x<π2.

LHS=cot−1(√1+sinx+√1−sinx√1+sinx−√1−sinx)sinx=2sinx2cosx2⇒cot−1(√(cosx2+sinx2)2+√(cosx2−sinx2)2√(cosx2+sinx2)2−√(cosx2−sinx2)2⇒cot−1(cosx2+sinx2+cosx2−sinx2cosx2+sinx2−cosx2+sinx2)⇒cot−1(2cosx22sinx2)⇒cot−1(cotx2)=x2=RHS

Prove that ${cot}^{- 1} {\frac{\sqrt{1 + sin x} + \sqrt{1 - sin x}}{\sqrt{1 + sin x} - \sqrt{1 - sin x}}} = \frac{x}{2}, 0 < x < \frac{π}{2} .$