Question

${ \tt \: if \tan \theta \: = \frac{1}{ \sqrt{2} }} \: \: , \\ \\ { \tt \: find \: the \: value \: of \: } \\ { \tt \: \frac{ \cosec^{2} \theta- { \sec}^{2} \theta }{ \cosec^{2} \theta + { \cot}^{2} \theta } }$
⚠︎ ᴅᴏɴ'ᴛ sᴘᴀᴍ
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Answer 1

Correct Question :-

$\sf If \ tan \theta = \dfrac{1}{\sqrt{2}}$,

Find the value of :-

$\sf \dfrac{cosec^2 \theta-sec^2 \theta }{cosec^2 \theta+cot^2 \theta}$

Solution :-

We know :-

$\sf cot \theta = \dfrac{1}{tan \theta }$

$\sf\therefore cot \theta = \sqrt{2}$

$\bullet \bf \ cosec^2 \theta = 1+cot^2 \theta \\\\\bullet \ sec^2 \theta = 1+tan^2 \theta$

$\longrightarrow \sf \dfrac{cosec^2 \theta-sec^2 \theta}{cosec^2 \theta+cot^2 \theta} \\\\\longrightarrow \dfrac{1+cot^2 \theta -( 1+tan^2 \theta) }{1+cot^2 \theta +cot^2 \theta} \\\\ \longrightarrow \dfrac{1+cot^2 \theta -1 - tan^2 \theta}{1+cot^2 \theta + cot^2 \theta} \\\\\longrightarrow \dfrac{cot^2 \theta - tan^2 \theta}{1+2 cot^2 \theta }$

Substitute the values of $\sf tan \theta$ and $\sf cot \theta$,

$\longrightarrow \sf \dfrac{ (\sqrt{2} ) ^2 - \left( \dfrac{1}{\sqrt{2}} \right)^2}{1+ 2 \times (\sqrt{2})^2} \\\\ \longrightarrow \dfrac{2- \dfrac{1}{2}}{1+ 2\times 2 } \\\\ \longrightarrow\dfrac{\dfrac{4-1}{2}}{1+4} \\\\ \longrightarrow \dfrac{\dfrac{3}{2}}{5} \\\\ \longrightarrow \dfrac{3}{2} \times \dfrac{1}{5} \\\\ \longrightarrow \bf \dfrac{3}{10}$

Answer 2

Step-by-step explanation:

Given :

• $\sf \: \tan \theta \: = \frac{1}{ \sqrt{2} } \: \:$

To Find :

• $\sf \: \frac{ \cosec^{2} \theta- { \sec}^{2} \theta }{ \cosec^{2} \theta + { \cot}^{2} \theta } \:$

Solution :

According to the Question :

$\sf : \implies\:\: \: \: \: \: \: \: \: \: \: \: \: \: \frac{ \cosec^{2} \theta- { \sec}^{2} \theta }{ \cosec^{2} \theta + { \cot}^{2} \theta } \:$

$\\ \\ \sf : \implies\:\: \: \: \: \: \: \: \: \: \: \: \: \: \frac{1 + {cot}^{2} - 1 - {tan}^{2} }{1 + {cot}^{2} + {cot}^{2} }$

$\sf : \implies\:\: \: \: \: \: \: \: \: \: \: \: \: \: \frac{ {cot}^{2} - {tan}^{2} }{2 {cot}^{2} + 1 } \\ \\ \\ \sf : \implies\:\: \: \: \: \: \: \: \: \: \: \: \: \: \frac{2 - \frac{1}{2} }{2 \times 2 + 1} \\ \\ \\ \sf : \implies\:\: \: \: \: \: \: \: \: \: \: \: \: \: \frac{ \frac{4 - 1}{2} }{5} \\ \\ \\ \sf : \implies\:\: \: \: \: \: \: \: \: \: \: \: \: \: \frac{ \frac{3}{2} }{5} \\ \\ \\ \sf : \implies\:\: \: \: \: \: \: \: \: \: \: \: \: \: \frac{3}{2} \times \frac{1}{5} \\ \\ \\ \sf : \implies\:\: \: \: \: \: \: \: \: \: \: \: \: \: \frac{3}{10}$

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$\boxed{\begin{minipage}{6cm} Important Trigonometric identities :- \\ \\ \: \: 1)\:\sin^2\theta+\cos^2\theta=1 \\ \\ 2)\:\sin^2\theta= 1-\cos^2\theta \\ \\ 3)\:\cos^2\theta=1-\sin^2\theta \\ \\ 4)\:1+\cot^2\theta=\text{cosec}^2 \, \theta \\ \\5)\: \text{cosec}^2 \, \theta-\cot^2\theta =1 \\ \\ 6)\:\text{cosec}^2 \, \theta= 1+\cot^2\theta \\\ \\ 7)\:\sec^2\theta=1+\tan^2\theta \\ \\ 8)\:\sec^2\theta-\tan^2\theta=1 \\ \\ 9)\:\tan^2\theta=\sec^2\theta-1 \end{minipage}}$