Differentiation of inverse functions using graphs with conditions?

Question

I was trying to differentiate this equation. And I got the answer but it matches none. Any help on how to solve this one. I tried by converting this function to $y=tan^{-1}tan{\frac{x}2} $ and then using condition to differentiate but my answer is not coming. Please help me solve this.The correct answer is option (b) Second question is I am startled about how can value of $ cos{\frac{x}2} $ & $ sin{\frac{x}2} $ affect the differentiation.

Answer 1

As you did before, by writing the fraction with respect to $k=\tan (x/2)$, we have $$y=\frac{\sqrt{1+2k/(1+k^2)}+\sqrt{1-2k/(1+k^2)}}{\sqrt{1+2k/(1+k^2)}-\sqrt{1-2k/(1+k^2)}}=\frac{|k+1|+|k-1|}{|k+1|-|k-1|}$$ So, if $k-1>0$ which is equivalent to $\sin(x/2)>\cos(x/2)$ we get $$y=\frac{k+1+k-1}{k+1-k+1}=k=\tan(x/2)$$ Now think about this latter one.