Identify whether the following equation is true or false

Identify whether the following equation is true or false: sin x + sin y = sin (x+y) Choose the correct answer below: A. False, because sin x + sin y = sin (x + y) cos (x – y) and sin (x + y) = sin x cos y + cos x sin y B. True, because sin x + sin y = sin x cos y + cos x sin y which is equal to sin (x+y) C. True, because sin x + sin y = 2 sin x cos x which is equal to sin (x + y) D. False, because sin x + sin y = 2 sin x cos x and sin (x + y) = sin x cos y + cos x sin y

The Correct Answer and Explanation is:

The correct answer is D: False, because sin x + sin y = 2 sin x cos x and sin (x + y) = sin x cos y + cos x sin y.

Explanation:

The equation given is sin x + sin y = sin (x + y), which is false. Let’s break it down:

1. The Left-Hand Side (sin x + sin y):
   The sum of the sines of two angles can be expressed using the sum-to-product identity in trigonometry: sin⁡x+sin⁡y=2sin⁡(x+y2)cos⁡(x−y2)\sin x + \sin y = 2 \sin\left(\frac{x + y}{2}\right) \cos\left(\frac{x – y}{2}\right)sinx+siny=2sin(2x+y​)cos(2x−y​) So, the sum of the sines is not simply equal to the sine of the sum of the angles. Instead, it involves both sine and cosine functions.
2. The Right-Hand Side (sin (x + y)):
   The formula for the sine of a sum of two angles is given by: sin⁡(x+y)=sin⁡xcos⁡y+cos⁡xsin⁡y\sin(x + y) = \sin x \cos y + \cos x \sin ysin(x+y)=sinxcosy+cosxsiny This is the sum formula for sine, which expresses the sine of a sum as the sum of products of sines and cosines.

Key Difference:

• sin x + sin y is not equal to sin(x + y).
• The identity for the sum of sines (as used on the left-hand side) involves both sine and cosine terms, while the identity for sin(x + y) involves products of sine and cosine of the individual angles.

Thus, the statement is false. The correct formula for sin(x + y) is sin x cos y + cos x sin y, which is very different from the sum of the two sines.