➪ 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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[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.
Refer to the attachment..........