Let $$A=sinx+tanx$$ and $$B=2x$$ in the interval $$0$$

Question

Let $$A=sinx+tanx$$ and $$B=2x$$ in  the  interval $$0$$

Infinity Learn · Accepted Answer

We have $$A=sinx+tanx$$, $$B=2xA$$−$$B=sinx+tanx$$−$$2x$$                     $$=$$ sinx−$$x+tanx$$−xLet $$f(x)=sinx$$−$$x+tanx$$−x⇒         $$f1(x)=cosx$$−$$1+sec2x$$−$$1$$                               $$=cosx$$−$$1+tan2x$$⇒$$f11(x)=$$−$$sinx+2tanxsec2x$$                             $$=tanx2sec2x$$−cosx>$$0$$ in $$0$$,π$$2$$⇒$$f1(x)$$ is increasing in $$0$$,π$$2$$             ∴x>$$0$$⇒$$f1x$$>$$f10$$               ⇒$$f1(x)$$>$$0$$           ⇒$$f(x)$$ is an increasing function in $$0$$,π$$2$$           ∴ x>$$0$$⇒$$f(x)$$>$$f(0)$$⇒                      $$f(x)$$>$$0$$ ⇒            A−B>$$0$$⇒A>B

$L e t A = \sin x + \tan x a n d B = 2 x i n t h e i n t e r v a l 0 < x < \frac{π}{2}$ then

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Let A=sinx+tanx and B=2x in the interval 0<x<π2then

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$L e t A = \sin x + \tan x a n d B = 2 x i n t h e i n t e r v a l 0 < x < \frac{π}{2}$ then