Question

If $a\cos 2\theta +b\sin 2\theta =c$ has $\alpha$ and $\beta$ as its roots then $\tan \alpha +\tan \beta =\frac{2b}{a+c}$.

State whether true or false.

• A. True
• B. False

Answer 1

The correct option is A True

$a\cos 2\theta +b\sin 2\theta =c$

$\cos 2\theta =\frac{1-{\tan }^2\theta }{1+{\tan }^2\theta }$

$\sin 2\theta =\frac{2\tan \theta }{1+{\tan }^2\theta }$

$a\left(\frac{1-{\tan }^2\theta }{1+{\tan }^2\theta }\right)+2b\frac{\tan \theta }{1+{\tan }^2\theta }=c$

$a(1-{\tan }^2\theta )+2b\tan \theta =c+c{\tan }^2\theta$

$=-{\tan }^2\theta (a+c)+2b\tan \theta +a-c=0$

$rootsbe\tan \alpha ,\tan \beta$

$\tan \alpha +\tan \beta =\frac{2b}{a+c}$