Question

If $\alpha ,\beta$ are the acute angles of an obtuse-angled triangle, then $\tan \alpha \tan \beta$ is

• A. $<1$
• B. $>1$
• C. $=1$
• D. None of these

Answer 1

The correct option is A

$<1$

Explanation for the correct option:

Find the value of $\tan \alpha \tan \beta$:

Given, $\alpha ,\beta$ are the acute angles in an obtuse angled triangle

$\Rightarrow$ $\alpha +\beta <90°$

$\Rightarrow$ $\tan \left(\alpha +\beta \right)>0$

$\Rightarrow$$\frac{\tan \alpha +\tan \beta }{1–\tan \alpha \tan \beta }>0$

$\Rightarrow$ $1–\tan \alpha \tan \beta >0$

$\Rightarrow$ $\mathit{t}\mathit{a}\mathit{n}\mathbf{}\mathit{\alpha }\mathbf{}\mathit{t}\mathit{a}\mathit{n}\mathbf{}\mathit{\beta }\mathbf{}\mathbf{<}\mathbf{}\mathbf{1}$

Hence, Option ‘A’ is Correct.