Question

cosec x -cot x=2,find cosec^2 x + cot ^2 x

Answer 1

$\cosec x - \cot x=2$

$\implies \frac{1}{\sin x} - \frac{\cos x}{ \sin x}=2$

$\implies \frac{1 - \cos x}{ \sin x}=2$

$\implies \frac{2 \sin^{2} (\frac{x}{2}) }{ 2\sin (\frac{x}{2}) . \cos (\frac{x}{2} )}=2$

$\implies \frac{\sin (\frac{x}{2}) }{ \cos (\frac{x}{2} )}=2$

$\therefore \tan( \frac{x}{2} ) =2$

$\blue{ \boxed{ \tan x = \frac{2 \tan( \frac{x}{2} ) }{1 - \tan ^{2} ( \frac{x}{2} ) } }}$

$\implies \tan x = \frac{2(2)}{1 - {(2)}^{2} }$

$\implies \tan x = \frac{4}{1 - 4}$

$\therefore \tan x = \frac{ - 4}{3}$

$\implies \cosec x = \frac{ -5}{4}$

$\implies \cot x = \frac{ - 3}{4}$

$\cosec^2 x + \cot ^2 x$

$\implies {( \frac{ - 5}{4} )}^{2} + {( \frac{ - 3}{4}) }^{2}$

$\implies {\frac{ 25}{16} } + { \frac{ 9}{16} }$

$\implies {\frac{ 25 + 9}{16} }$

$\implies {\frac{ 34}{16} }$

$\orange{\therefore \frac{17}{8}}$

HOPE IT HELPS!!