sin (x) + cos (x) = 0. Why this equation has only one solution set?

Question

The equation "sin (x) + cos (x) = 0" has only one solution set "$x=\frac{3\pi }{4}+\pi n$".

Why it has not solution set "$x=\frac{7\pi }{4}+\pi n$"? Although it satisfy the equation.

Please help quickly.

Answer 1

The equation is equivalent to $$\tan x=-1$$ since the two functions $\cos$ and $\sin$ don't vanish together so we find $$x\equiv\frac{3\pi}4\mod \pi$$

Answer 2

A solution set is a set of points that satisfies a given equation. A given equation will have only one solution set. That set can have many descriptions. $\frac {3\pi}4+n\pi$ is one description of the solution set for this equation. $\frac {7\pi}4+m\pi$ is another description of the same set.

Answer 3

Note that $$\frac{7\pi}{4} + n\pi = \left(1 + \frac34\right)\pi + n\pi = \frac{3\pi}{4} + (n+1)\pi,$$ so you are naming the same set of solutions but with a different indexing system.