Question

if y=log tan x/2 find dy/dx

Answer 1

Probably this is the answer

Answer 2

Answer:

$\frac{\mathrm{d}y}{\mathrm{d}x}=cosec\,x}$

Step-by-step explanation:

Given: y = log( tan(x/2) )

We have to find: dy/dx

Consider,

$\frac{\mathrm{d}y}{\mathrm{d}x}=\frac{\mathrm{d}(log\,(tan\,\frac{x}{2}))}{\mathrm{d}x}$

$=\frac{1}{tan\frac{x}{2}}\times\frac{\mathrm{d}(tan\,\frac{x}{2})}{\mathrm{d}x}$

$=\frac{1}{tan\,\frac{x}{2}}\times sec^2\,\frac{x}{2}\times\frac{\mathrm{d}(\frac{x}{2})}{\mathrm{d}x}$

$=\frac{cos\,\frac{x}{2}}{sin\,\frac{x}{2}}\times\frac{1}{cos^2\,\frac{x}{2}}\times\frac{1}{2}$

$=\frac{1}{sin\,\frac{x}{2}}\times\frac{1}{cos\,\frac{x}{2}}\times\frac{1}{2}$

$=\frac{1}{2\times sin\,\frac{x}{2}\times cos\,\frac{x}{2}}$

$=\frac{1}{sin\,(2\times\frac{x}{2})}$

$=\frac{1}{sin\,x}$

$=cosec\,x}$

Therefore, $\frac{\mathrm{d}y}{\mathrm{d}x}=cosec\,x}$