If y=tan−1⁡acos⁡x−bsin⁡xbcos⁡x+asin⁡x where −π2 −1 then⁡−dydx2=

$$y=$$$$tan$$−$$1$$⁡ab−$$tan$$⁡$$x1+abtan$$⁡$$x=$$$$tan$$−$$1$$⁡ab−$$tan$$−$$1$$⁡$$($$$$tan$$⁡$$x)$$⇒$$dydx=0$$−$$1=$$−$$1$$

Q.

If $y = \tan^{- 1} (\frac{a \cos x - b \sin x}{b \cos x + a \sin x})$ where $- \frac{π}{2} < x < \frac{π}{2} and \frac{a}{b} \tan x > - 1$ $then \frac{(\frac{- d y}{d x})}{2} =$

Moderate

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By Expert Faculty of Sri Chaitanya

answer is 0.5.

(Detailed Solution Below)

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Detailed Solution

$y = \tan^{- 1} (\frac{\frac{a}{b} - \tan x}{1 + \frac{a}{b} \tan x}) = \tan^{- 1} (\frac{a}{b}) - \tan^{- 1} (\tan x)$

$\Rightarrow \frac{d y}{d x} = 0 - 1 = - 1$

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