If \(\cos x+\cos y=0\) and \(\sin x+\sin y=0\) then \(\cos (x-y)=\) (a) 1 (b) \((1 / 2)\) (c) \(-1\) (d) \(-(1 / 2)\)

Understanding the cosine sum formula is essential for solving various trigonometric problems including the one where we need to determine the value of \(\cos(x-y)\) when given \(\cos x + \cos y = 0\) and \(\sin x + \sin y = 0\). The cosine sum formula is given by \(\cos(\alpha + \beta) = \cos \alpha \cos \beta - \sin \alpha \sin \beta\). This formula tells us how to express the cosine of the sum of two angles in terms of the sines and cosines of the individual angles.
We also have a similar formula for the difference of two angles, \(\cos(\alpha - \beta) = \cos \alpha \cos \beta + \sin \alpha \sin \beta\), which was used in the provided step-by-step solution. By substituting the values found for \(\cos y\) and \(\sin y\) into the \(\cos(x-y)\) expression and simplifying, we utilize this cosine difference formula to derive the solution to the exercise.
Essentially, the cosine sum formula is a building block for transforming and simplifying trigonometric expressions and is a powerful tool for proving various other trigonometric identities. When this formula is committed to memory, it becomes much easier to approach and solve complex trigonometric equations.

The sine function, denoted as \(\sin\), is a fundamental trigonometric function that represents the ratio of the length of the opposite side to the length of the hypotenuse in a right-angled triangle. It's an essential component in the study of trigonometry and appears in a variety of mathematical contexts.
In the context of our problem, we had the equation \(\sin x + \sin y = 0\). This implies that the sine values of \(x\) and \(y\) are negatives of each other. The sine function has a range of \( -1 \leq \sin \theta \leq 1\), which means it can take on any value within that interval. It is also worth noting that the sine function is periodic, with a period of \(2\pi\), meaning that \(\sin(\theta + 2\pi) = \sin \theta\).
The sine function is related to the cosine function through the phase shift property, which states that \(\sin(\theta) = \cos(\theta - \pi/2)\). This property, along with the sine function's behavior, helps us understand and solve equations involving trigonometric functions.

The Pythagorean identity is one of the most well-known and widely used identities in trigonometry. It links the sine and cosine functions in a simple yet profound equation: \(\sin^2 \theta + \cos^2 \theta = 1\).
This identity is derived from the Pythagorean theorem, a fundamental principle in geometry involving right-angled triangles. The equation states that the square of the hypotenuse is equal to the sum of the squares of the other two sides. In trigonometric terms, since the sine and cosine values represent the lengths of sides in a unit circle, their squares add up to the radius of the circle squared, which is 1.
In our exercise, the Pythagorean identity was crucial in step 7 to simplify the expression \(\cos(x-y) = -\cos^2 x - \sin^2 x\) to \(\cos(x-y) = -1\) by recognizing that \(\sin^2 x + \cos^2 x = 1\). Without the use of this identity, the simplification and thus the solution to the problem would be much more complex. By mastering the Pythagorean identity, students gain a powerful tool for simplifying trigonometric expressions and solving equations.