Question

If y= log(tan pi/4+x/2) prove that dy/dx=sec x

Answer 1

Hence proved!
2sinAcosA=sin2A
tan=sin/cos

Answer 2

Answer:

Given,

$y = logtan( \frac{\pi}{4} + \frac{x}{2})$

We are differentiating y with respect to x.

$\frac{dy}{dx} = \frac{1}{tan( \frac{\pi}{4} + \frac{x}{2}) } \times \frac{d}{dx} tan( \frac{\pi}{4} + \frac{x}{2}) \\ \frac{dy}{dx} = \frac{1}{tan( \frac{\pi}{4} + \frac{x}{2}) } \times {sec}^{2} ( \frac{\pi}{4} + \frac{x}{2} ) \times \frac{d}{dx} ( \frac{x}{2} ) \\ \frac{dy}{dx} = \frac{1}{tan( \frac{\pi}{4} + \frac{x}{2}) } \times {sec}^{2} ( \frac{\pi}{4} + \frac{x}{2} ) \times \frac{1}{2}$

$\frac{dy}{dx} = cot( \frac{\pi}{4} + \frac{x}{2}) \times {sec}^{2} ( \frac{\pi}{4} + \frac{x}{2}) \times \frac{1}{2}$

$\frac{dy}{dx} = \frac{ \cos( \frac{\pi}{4} + \frac{x}{2}) }{ \sin(\frac{\pi}{4} + \frac{x}{2})} \times \frac{1}{ {cos}^{2} ( \frac{\pi}{4} + \frac{x}{2})} \times \frac{1}{2}$

$\frac{dy}{dx} = \frac{1}{2sin( \frac{\pi}{4} + \frac{x}{2})cos( \frac{\pi}{4} + \frac{x}{2})}$

$\frac{dy}{dx} = \frac{1}{sin2( \frac{\pi}{4} + \frac{x}{2})} \\ = \frac{1}{sin( \frac{\pi}{2} + x)} \\ = \frac{1}{cosx}$

Therefore

$\frac{dy}{dx} = \sec(x)$

Hence proved.

Here applied formulas are

$\tan(\theta) = \frac{ \sin(\theta) }{ \cos(\theta) } \\ \cos(\theta) = \frac{1}{ \sec(\theta) } \\ \sin(2\theta) = 2 \sin(\theta) \cos(\theta) \\ \frac{d}{dx} ( log(a)) = \frac{1}{a} \\ \frac{d}{dx} ( \tan(\theta) ) = {sec}^{2} \theta \\ sin(\pi + \theta) = \cos(\theta)$

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