Fundamental law of trigonometry Formula: Let Alpha(α) and beta (β) any two angle (real number),
then
cos (α – β) = cos α cos β + sin α sin β Which is called Fundamental law of trigonometry Formula.
PROOF:
we sages that Alpha(α) greater than beta (β) and beta greater than zero α > β> 0
Consider a unit circle with center at origin
Let Terminal sides of angle α and β cut the unit circle at A and B respectively
First terminal ray Angle ∠ AOB = α – β
And
Take a point C on the unit circle
so that
Angle ∠ X O C = Angle ∠ AOB = α – β
join A, B and C, D
Now angles Alpha(α) and beta (β) and α – β are in the standards position.
Therefore,
the coordinate of A arc (cos α, sin α)
The coordinate of Arc B (cos β, sin β)
The coordinate of Arc C are (cos α – β, sin α – β)
and
The coordinate of Arc D (1, 0)
Now Δ AOB and Δ COD are congruent.
Therefore,
|AB| = |CD| this implies that |AB|² = |CD|²
Using the distance formula
(Cos α – Cos β)²+(Sin α – Sin β)²= [Cos (α – β)-1]²+[Sin(α – β) – 0]²
this implies that
Cos²α + Cos²β-2Cos α Cos β + Sin² α Sin² β – 2 Sin α sin β
this is equal to
Cos² (α – β) +1 -2 Cos (α -β) + Sin²(α −β)
This is equal to
2 – 2 (Cos α Cos β+ Sin α Sin β) = 2 – 2 Cos (α – β)
After cancelation, we get the result
cos (α – β) = cos α cos β + sin α sin β
When Alpha(α) greater than beta (β) greater than zero then this formula is true for all values of Alpha(α) and beta (β)
α >β> 0