Question

No. 13
Find the value of cot θ - cos θ
If cosec θ= √5

Answer 1

Given, cosec $\theta$ = $\sqrt{5}$

$\textsf{\underline {By using the trigonometric identity}}$,

$\fbox{\tt{ \sqrt{1\:+\:cosec^{2}\:\theta}}\:= \: cot\:\theta}$

$\sqrt{\tt{1\:+\:\sqrt{(5)}^{2}\:\theta}}$ = $cot\:\theta$

➡️ $\fbox{\tt{ cot\:\theta \: =\: \sqrt{6}}}$ --> ( i )

Now,

$\fbox{ \tt{\frac{1}{cosec\:\theta}}}$ = $\tt{sin\:\theta}$

➡️ $\tt{sin\:\theta}$ = $\fbox{ \tt{\frac{1}{\sqrt{5}}}}$

$\textsf{\underline {By using the trigonometric identity}}$,

$\fbox{\tt{\sqrt{1\:-\:sin^{2}\:\theta}}\: =\: cos\:\theta}$

$\tt{\sqrt{1\:-\:(\frac{1}{\sqrt{5}}) ^{2}}\: =\: cos\:\theta}$

➡️ $\tt{\sqrt{\frac{4}{5}}} \:=\:cos\:\theta$ --> ( ii )

$\sf{\underline {Value \:of\:( cot\:\theta\:-\:cos\:\theta) }}$ :

Putting value from ( i ) and ( ii ),

➡️ $\tt{cot\:\theta\:-\:cos\:\theta\:=\:\sqrt{6}\:-\:\sqrt{\frac{4}{5}}}$