In a circle with radius=r,take 2 points P and Q such that P is at an angle x and Q at an angle y
the co-ordinates of P and Q are P(cosx,sinx),Q(cosy,siny)
distance between P and Q is:
(by distance formula)
(PQ)²=(cosx−cosy)²+(sinx−siny)²=2−2(cosx.cosy+sinx.siny)
(PQ)²=1²+1²−2cos(x−y)=2−2cos(x−y)
cos(x−y)=cos(x)cos(y)+sin(x)sin(y)
Substituting y with −y we get
cos(x+y)=cosxcosy−sinxsiny

Proof:

Introduction:
We need to prove geometrically that Cos(x y) = Cos(x) * Cos(y) - Sin(x) * Sin(y).

Step 1: Definitions:
Let's define the trigonometric functions involved in this equation:
- Cos(x): The cosine of angle x is the ratio of the adjacent side to the hypotenuse in a right triangle.
- Sin(x): The sine of angle x is the ratio of the opposite side to the hypotenuse in a right triangle.
- Cos(x y): The cosine of the sum of angles x and y.

Step 2: Consider a Right Triangle:
Consider a right triangle ABC, where angle A = x and angle B = y. Let AC be the hypotenuse and BC be the adjacent side to angle A. Let AB be the opposite side to angle A.

Step 3: Use Trigonometric Ratios:
Using trigonometric ratios in triangle ABC, we have:
- Cos(x) = BC / AC
- Sin(x) = AB / AC
- Cos(y) = AC / BC
- Sin(y) = AB / BC

Step 4: Apply the Sum of Angles Formula:
According to the sum of angles formula for cosine, we have:
Cos(x + y) = Cos(x) * Cos(y) - Sin(x) * Sin(y)

Step 5: Substitute the Values:
Substituting the values from Step 3 into the equation from Step 4, we get:
Cos(x + y) = (BC / AC) * (AC / BC) - (AB / AC) * (AB / BC)

Step 6: Simplify:
Simplifying the equation, we get:
Cos(x + y) = 1 - (AB / AC)^2

Step 7: Apply Pythagorean Identity:
Using the Pythagorean identity Sin^2(x) + Cos^2(x) = 1, we have:
(AB / AC)^2 + (BC / AC)^2 = 1

Step 8: Simplify:
Simplifying the equation, we get:
(AB)^2 + (BC)^2 = (AC)^2

Step 9: Use the Pythagorean Theorem:
The equation (AB)^2 + (BC)^2 = (AC)^2 represents the Pythagorean theorem, which states that the sum of the squares of the lengths of the two legs of a right triangle is equal to the square of the length of the hypotenuse.

Conclusion:
Therefore, using geometric reasoning and the definitions of trigonometric functions, we have proven that Cos(x y) = Cos(x) * Cos(y) - Sin(x) * Sin(y).