TITLE: Trigonometric ratio of multiple and sub multiple angles
QUESTION [5 upvotes]: Given that $a$ lies in 1st quadrant and 
$$ \sin a +\cos a +\operatorname{cosec} a+\sec a+\tan a+\cot a=7$$
then we have to prove that $\sin(2a)$ is a root of $$x^2-44x-36.$$ I have tried to break all of them in respect of sine and cosines but that only makes my problem complex. I just cannot relate to the root part. Can anyone help?

REPLY [10 votes]: Combine $\sin a$ and $\csc a$, $\cos a$ and $\sec a$, and $\tan a$ and $\cot a$. And convert all the terms in $\sin$ and $ \cos$. Then cross multiply. You should get:$$\sin a\cos a(\sin a+\cos a)+(\sin a+\cos a)+1-7\sin a\cos a=0$$
Substitute $\sin a+\cos a=t$, you have $\sin a\cos a=\frac{t^2-1}{2}$, you get a cubic:
$$t^3-7t^2+t+9=0$$$$(t+1)(t^2-8t+9)=0$$
You will have to reject the $t=-1$ case.(You can figure that out). 
Taking the root of the second equation you get:
$\sin a+\cos a=4-\sqrt 7$. Squaring both sides, you get:
$$1+\sin 2a=23-8\sqrt 7$$$$\sin 2a=22-8\sqrt 7$$ And therefore the other root of a quadratic with rational coefficients would be $22+8\sqrt 7$, and hence the result follows that $\sin 2a$ will be a root of $x^2-44x+36=0$