First slide

Multiple and sub- multiple Angles

Question

If $\tan \frac{α}{2} and \tan \frac{β}{2}$ are the roots of the equation $8 x^{2} - 26 x + 15 = 0, then \cos (α + β) =$

Moderate

A

$- \frac{627}{725}$
B

$\frac{627}{725}$
C

-1
D

none of these

Solution

We have,
$\tan \frac{α}{2} + \tan \frac{β}{2} = \frac{13}{4} and \tan \frac{α}{2} \tan \frac{β}{2} = \frac{15}{8}$
$\tan (\frac{α}{2} + \frac{β}{2}) = \frac{\tan \frac{α}{2} + \tan \frac{β}{2}}{1 - \tan \frac{α}{2} \tan \frac{β}{2}} = \frac{26}{7}$
Now,
$\cos (α + β) = \frac{1 - \tan^{2} (\frac{α + β}{2})}{1 + \tan^{2} (\frac{α + β}{2})} = \frac{1 - \frac{676}{19}}{1 + \frac{676}{49}} = - \frac{627}{725}$

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