Question

If $\tan \frac{\alpha }{2}$ and $\tan \frac{\beta }{2}$ are the roots of the equation $8x^2-26x+15=0$ then $\cos (\alpha +\beta )$ is equal to

• A. $-\frac{627}{725}$
• B. $\frac{627}{725}$
• C. $-1$
• D. $Noneofthese$

Answer 1

The correct option is A $-\frac{627}{725}$

As $\tan \frac{\alpha }{2},\tan \frac{\beta }{2}$ is root of $8x^2-26x+15=0$

Then, $\tan \frac{\alpha }{2}+\tan \frac{\beta }{2}=\frac{13}{4}$

and $\tan \frac{\alpha }{2}\cdot \tan \frac{\beta }{2}=\frac{15}{8}$

Now, $\tan \left(\frac{\alpha +\beta }{2}\right)=\frac{\tan \frac{\alpha }{2}+\tan \frac{\beta }{2}}{1-\tan \frac{\alpha }{2}\cdot \tan \frac{\beta }{2}}=\frac{\frac{13}{4}}{1-\frac{15}{8}}=-\frac{26}{7}$

Hence $\cos (\alpha +\beta )=\frac{1-{\tan }^2\left(\frac{\alpha +\beta }{2}\right)}{1+{\tan }^2\left(\frac{\alpha +\beta }{2}\right)}=-\frac{627}{725}$