If $\sin A=\frac{1}{2}, A$ is an acute angle, then find the value of $\frac{\tan A-\cot A}{\sqrt{3}(1+{cosec} A)}$.

$-\frac{2}{9}$

We are given that :-

sinA = \(\frac{1}{2}\)

{ we know , sin 30º = \(\frac{1}{2}\) }

So, A = 30º

Now,

\(\frac{tanA - cotA}{ √3 ( 1 + cosecA ) }\)

= \(\frac{tan 30º - cot 30º}{ √3 ( 1 + cosec 30º ) }\)

= \(\frac{ 1/ √3 - √3}{ √3 ( 1 + 2 ) }\)

= \(\frac{ -2/ √3}{ 3√3  }\)

= \(\frac{ -2}{ 9  }\)