➪ Find value of ;
[tex]{ \bold{ \tt(1 + \tan \theta \: + sec \theta)(1 + \cot \theta \: - \cosec \theta)}}[/tex]
⚠︎ ᴅᴏɴ'ᴛ sᴘᴀᴍ
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➪ Find value of ;
[tex]{ \bold{ \tt(1 + \tan \theta \: + sec \theta)(1 + \cot \theta \: - \cosec \theta)}}[/tex]
⚠︎ ᴅᴏɴ'ᴛ sᴘᴀᴍ
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Answer :-
( 1 + tan theta + sec theta ) ( 1 + cot theta - Cosec theta)
= ( 1 + sin theta/ cos theta + 1/cos theta) ( 1 + cos theta / sin theta - 1 / sin theta)
By taking LCM ,
= ( cos theta + sin theta + 1 ) ( sin theta + cos theta - 1) / cos theta * sin theta
= ( cos theta + sin theta )^2 - 1 / cos theta * sin theta
By using identity ( a + b) ^2 = a^2 + b^2 + 2ab
= cos^2 theta + sin^2 theta + 2 * cos theta * sin theta - 1 / cos theta * sin theta
As we know, cos^2 theta + sin^2 theta = 1
= 1 + 2 * cos theta * sin theta - 1 / cos theta * sin theta
= 1 - 1 * 2 * cos theta * sin theta / cos theta * sin theta
= 2 * cos theta * sin theta / cos theta * sin theta
= 2 is your answer mate
Some more info :-
Verified answer
Answer:-
[tex]\red{\bigstar}[/tex] Value is [tex]\large\underline{\boxed{\tt\purple{2}}}[/tex]
• Given:-
[tex]\sf(1 + tan \theta \: + sec \theta)(1 + cot \theta \: - cosec \theta)[/tex]
★ Basic Knowledge:-
➪ [tex]\sf\pink{tan \theta = \dfrac{sin \theta}{cos \theta}}[/tex]
➪ [tex]\sf\pink{cot \theta = \dfrac{cos \theta}{sin \theta}}[/tex]
➪ [tex]\sf\pink{sec \theta = \dfrac{1}{cos \theta}}[/tex]
➪ [tex]\sf\pink{cosec \theta = \dfrac{1}{sin \theta}}[/tex]
➪ [tex]\sf\pink{sin^2 \theta + cos^2 \theta = 1}[/tex]
• Solution:-
➪ [tex]\sf(1 + tan \theta \: + sec \theta)(1 + cot \theta \: - cosec \theta)[/tex]
➪ [tex]\sf \bigg(1 + \dfrac{sin \theta}{cos \theta}+ \dfrac{1}{cos \theta} \bigg) \bigg(1+ \dfrac{cos \theta}{sin \theta} - \dfrac{1}{sin \theta} \bigg) \\ [/tex]
➪ [tex]\sf \bigg(\dfrac{cos \theta + sin \theta + 1 }{cos \theta} \bigg) \bigg(\dfrac{sin \theta+ cos \theta - 1}{sin \theta} \bigg) \\ [/tex]
➪ [tex]\sf \bigg(\dfrac{(sin \theta + cos \theta - 1)(cos \theta + sin \theta + 1)}{cos \theta . sin \theta} \bigg) \\ [/tex]
➪ [tex]\sf \bigg(\dfrac{(sin \theta + cos \theta - 1)(sin \theta + cos \theta + 1)}{sin \theta . cos \theta} \bigg) \\ [/tex]
➪ [tex]\sf \bigg(\dfrac{(sin \theta + cos \theta)^2 - 1^2}{sin \theta . cos \theta} \bigg) \\ [/tex]
➪ [tex]\sf \bigg(\dfrac{sin^2 \theta + cos^2 \theta+ 2 sin cos \theta - 1}{sin \theta . cos \theta} \bigg) \\ [/tex]
➪ [tex]\sf \bigg(\dfrac{1 + 2 sin cos \theta - 1}{sin \theta . cos \theta} \bigg) \\ [/tex]
➪ [tex]\sf \bigg(\dfrac{2 sin \theta. cos \theta}{sin \theta . cos \theta} \bigg) \\ [/tex]
✯ [tex]\large{\bf\green{2}} \\ [/tex]
Therefore, the value of [tex]\sf(1 + tan \theta \: + sec \theta)(1 + cot \theta \: - cosec \theta)[/tex]
is 2.