How cos(x+y)=Cosx.cosy-sinx.siny?

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Answer

Answer

Derivation for formula of cos (A +B)

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Answer

Explanation:

To prove the identity cos(x + y) = cos(x)cos(y) - sin(x)sin(y), we will use the trigonometric properties of cosine and sine functions.

Step 1: Deconstructing cos(x + y)
To understand the identity, let's start by expanding cos(x + y) using the angle addition formula for cosine:

cos(x + y) = cos(x)cos(y) - sin(x)sin(y)

Step 2: Understanding the Angle Addition Formula for Cosine
The angle addition formula for cosine states that the cosine of the sum of two angles is equal to the product of the cosines of the individual angles minus the product of the sines of the individual angles.

Step 3: Breaking Down the Formula
Now, let's break down the formula to understand it better:

cos(x)cos(y) - sin(x)sin(y)

a) Product of cosines:
The product of cos(x) and cos(y) represents the cosine of the angles x and y multiplied together. This term focuses on the horizontal components of the angles.

b) Product of sines:
The product of sin(x) and sin(y) represents the sine of the angles x and y multiplied together. This term focuses on the vertical components of the angles.

c) Subtracting the vertical and horizontal components:
By subtracting the vertical components (product of sines) from the horizontal components (product of cosines), we obtain the final result of cos(x + y).

Step 4: Visualizing the Geometric Interpretation
Alternatively, we can visualize this identity geometrically using a unit circle. The cosine of an angle represents the x-coordinate of a point on the unit circle, while the sine represents the y-coordinate.

a) cos(x):
cos(x) represents the x-coordinate of a point on the unit circle corresponding to the angle x.

b) sin(x):
sin(x) represents the y-coordinate of a point on the unit circle corresponding to the angle x.

c) cos(x)cos(y):
cos(x)cos(y) represents the product of the x-coordinates of two points on the unit circle corresponding to angles x and y.

d) sin(x)sin(y):
sin(x)sin(y) represents the product of the y-coordinates of two points on the unit circle corresponding to angles x and y.

e) Subtracting the y-components from the x-components:
By subtracting the y-components (sin(x)sin(y)) from the x-components (cos(x)cos(y)), we obtain the x-coordinate of a point on the unit circle corresponding to the angle x + y. This is equivalent to cos(x + y).

Therefore, cos(x + y) = cos(x)cos(y) - sin(x)sin(y) holds true based on the angle addition formula for cosine and the geometric interpretation on the unit circle.

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