I'm trying to solve an equation here but unfortunately I can't.
The equation:
$$
\cos x + \sin x = 0
$$
I'm trying to solve this by replacing $\cos x$ with $(1-t^2)/(1+t^2)$ and $\sin x$ with $2t/(1+t^2), t=\tan x/2, \ $ but I can't get the right solution.
Also I have tried by squaring both sides but still nothing.
Can anyone help me ?
Best Answer
Note that $$\cos x + \sin x = 0 \iff \cos x = -\sin x$$
Now, $\cos x$ cannot equal zero, since if it did, $\sin x = -1$ or $\sin x = 1$, in which case the given equation isn't satisfied.
So we can divide by $\cos x$ to get $$1 = \dfrac{-\sin x}{\cos x} = -\tan x \iff \tan x = -1$$
Solving for $x$ gives us the values $x = \dfrac {3\pi}4 + k\pi$, where $k$ is any integer.