How to solve this trigonometric problem?

Question

I'm faced with this trigonometric problem:

$x$ and $y$ represent two angles in standard position. $x$ has its terminal arm in the first quadrant and $y$ has its terminal arm in the third quadrant.

If $\cos x = \frac 5{13}$ and $\cos y = \frac{-5}{13}$, find the value of:

$$2\sin x + 2\sin y + 2\cos x - 2\tan x + \tan y + 2\cos y$$

Aside from using a calculator, what ways are there to tackle this problem? Looking at it I epxect there to be some way to simplify/rearrange the equation to make it be only expressed in integers and cosines, because then all that's left is to switch the cosines with the fractions and solve the equation.

But I don't see any immediate way of doing that, at least not from the trig identities I know. Does anyone else see how it's supposed to be solved?

Answer 1

Since $\cos x=-\cos y$ and the two angles are in opposite quadrants, they must be opposite to each other, and $\sin x=-\sin y$, and $\tan x=\tan y$. Therefore $2\sin x+2\sin y$ and $2\cos x+2\cos y$ each cancel out and only $-\tan x$ is left. That can be dealt with quickly using Ross's hint.

Answer 2

Keep in mind that $$\tan\theta=\frac{\sin\theta}{\cos\theta},$$ and that $$\sin\theta=\pm\sqrt{1-\cos^2\theta},$$ where the sign is determined by the quadrant in which the terminal arm of $\theta$ lies.

Answer 3

Hint: $5^2+12^2=13^2$, so you can find the sines of the angles.