Question
Download Solution PDFযদি α এবং β ধনাত্মক কোণ হয় যেমন \(α + β = \dfrac{\pi}{4}\) , তাহলে (1 + tan α) (1 + tan β) এর মান কত?
Answer (Detailed Solution Below)
Detailed Solution
Download Solution PDFধারণা:
\(\rm \tan (α + β) = \dfrac {tan \alpha + tan \beta }{1 -tan \alpha \;tan \beta }\)
গণনা:
প্রদত্ত, α এবং β হল ধনাত্মক কোণ যেমন \(α + β = \dfrac{\pi}{4}\) ,
⇒ \(\rm \tan (α + β) = \tan (\dfrac{\pi}{4})\)
⇒ \(\rm \dfrac {tan \alpha + tan \beta }{1 -tan \alpha \;tan \beta } = 1\)
⇒ \(\rm {tan \alpha + tan \beta }= 1 -tan \alpha \;tan \beta \)
⇒ \(\rm {tan \alpha +tan \alpha \;tan \beta + tan \beta } - 1 = 0 \)
⇒ \(\rm {tan \alpha +tan \alpha \;tan \beta + tan \beta } +1-2= 0 \)
⇒ \(\rm {tan \alpha +tan \alpha \;tan \beta + tan \beta } +1=2\)
⇒ \(\rm tan \alpha (1 +tan \beta ) + (1+tan \beta)=2\)
⇒ \(\rm (1+ tan \alpha )(1 +tan \beta )=2\)
সুতরাং, α এবং β যদি ধনাত্মক কোণ হয় যেমন \(α + β = \dfrac{\pi}{4}\) , তাহলে (1 + tan α) (1 + tan β) 2 এর সমান।
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