Question

If $ \alpha +\beta =\frac{\pi }{4}, $ then the value of $ (1+\tan \alpha ) $ $ (1+\tan \beta ) $ is equal to

• A. $ 1 $
• B. $ -1 $
• C. $ 2 $
• D. $ -2 $

Answer 1

Answer: (C)

Given, $ \alpha +\beta =\frac{\pi }{4} $
$ \therefore $ $ \tan (\alpha +\beta )=\tan \left( \frac{\pi }{4} \right)=1 $
$ \Rightarrow $ $ \tan \alpha +\tan \beta =1-\tan \alpha \,\,tan\,\beta $
$ \Rightarrow $ $ \tan \,\,\alpha +\tan \,\,\beta +\tan \,\alpha \,\tan \,\beta =1 $ ..(i)
Now, $ (1+\tan \alpha )\,(1+\tan \beta ) $
$ =1+\tan \alpha +\tan \beta +\tan \alpha \,\tan \beta $
$ =1+1=2 $ [using Eq. (i)]