Differentiate given problems w-r-t-x- - 1-1-sin x-1-sin x-1-sin x-1-sin x- 0-x-2

Let y = cot^ 1 [ √(1+sin x)=√(1 sin x)/√(1+sin x) √(1 sin x) ],. =[ ( sinx/2+cosx/2 )+ ( cosx/2 sinx/2 )/ ( sinx/2+cosx/2 ) ( cosx/2 sinx/2 ) ] [ ∵ 1+sin x= ...

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Byju's Answer

Standard XII

Mathematics

Trigonometric Ratios of Multiples of an Angle

Differentiate...

Question

Differentiate given problems w.r.t.x.

$c o t^{-} 1 [\frac{\sqrt{1 + s i n x} + \sqrt{1 - s i n x}}{\sqrt{1 + s i n x} - \sqrt{1 - s i n x}}], 0 < x < \frac{π}{2}$ .

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Solution

Let y = $c o t^{-} 1 [\frac{\sqrt{1 + s i n x} = \sqrt{1 - s i n x}}{\sqrt{1 + s i n x} - \sqrt{1 - s i n x}}],$ .

= $[\frac{(s i n \frac{x}{2} + c o s \frac{x}{2}) + (c o s \frac{x}{2} - s i n \frac{x}{2})}{(s i n \frac{x}{2} + c o s \frac{x}{2}) - (c o s \frac{x}{2} - s i n \frac{x}{2})}]$

$⎡ ⎢ ⎢ ⎣ \begin{matrix} ∵ 1 + s i n x = s i n^{2} \frac{x}{2} + c o s^{2} \frac{x}{2} + 2 s i n \frac{x}{2} . c o s \frac{x}{2} = {(s i n \frac{x}{2} + c o s \frac{x}{2})}^{2} a n d 1 - s i n x = s i n^{2} \frac{x}{2} + c o s^{2} \frac{x}{2} - 2 s i n \frac{x}{2} c o s \frac{x}{2} = {(c o s \frac{x}{2} - s i n \frac{x}{2})}^{2} \end{matrix} ⎤ ⎥ ⎥ ⎦$

= $c o t^{-} 1 (\frac{2 c o s (x / 2)}{2 s i n (x / 2)}) = c o t^{1} (c o t (\frac{x}{2})) = \frac{x}{2} \Rightarrow \frac{d y}{d x} = \frac{1}{2}$

$I f \frac{π}{2} < X < π, t h e n \sqrt{1 - s i n x} = s i n \frac{x}{2} - c o s \frac{x}{2}$ .

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Differentiate given problems w.r.t.x. cot−1[√1+sin x+√1−sin x√1+sin x−√1−sin x],0<x<π2.

Differentiate given problems w.r.t.x.

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