Proof of Sum and Difference Identities
To demonstrate, the trigonometric sum and difference formulas let us consider a unit circle, with coordinates given as (cos θ, sin θ). Consider points A and B, which form angles of α and β with the positive X-axis, respectively. The coordinates of A and B are (cos α, sin α) and (cos β, sin β), respectively. We can observe that the angle AOB is equal to (α – β). Now, consider another two points P and Q on the unit circle such that Q is a point on the X-axis with coordinates (1,0) and angle POQ is equal to (α – β), and thus the coordinates of the point P are (cos (α – β), sin (α – β)).
Now, OA = OP, and OB = OQ as they are the radii of the same unit circle, and also the measure of one of the included angles of both triangles is (α – β).
Hence, by the side-angle-side congruence, triangles AOB and triangle POQ are congruent.
We know that the corresponding parts of congruent triangles are congruent, hence AB = PQ.
So, AB = PQ.
Using the distance formula between two points we get,
dAB = √[(cos α – cos β)2 + (sin α – sin β)2]
= √[cos2 α – 2 cos α cos β + cos2 β + sin2 α – 2 sin α sin β + sin2 β] {Since, (a – b)2 = a2 – 2ab + b2)}
= √[(cos2 α+ sin2 α) + (cos2 β+ sin2 β) – 2(cos α cos β + sin α sin β)]
= √[1 + 1 – 2(cos α cos β + sin α sin β)] {Since, sin2 x + cos2 x = 1}
= √[2 – 2(cos α cos β+ sin α sin β)]…….(1)
dPQ = √[(cos (α – β) – 1)2 + (sin (α – β) – 0)2]
= √[cos2 (α – β) – 2 cos (α – β) + 1 + sin2 (α – β)] {Since, (a – b)2 = a2 – 2ab + b2)}
= √[(cos2 (α – β) + sin2 (α – β)) + 1 – 2 cos (α – β)]
= √[1 + 1 – 2 cos (α – β)] {Since, sin2 x + cos2 x = 1}
= √[2 – 2 cos (α – β)]……(2)
Since AB = PQ, equate both equations (1) and (2).
√[2 – 2(cos α cos β+ sin α sin β)] = √[2 – 2 cos (α – β)]
By squaring on both sides, we get,
2 – 2(cos α cos β+ sin α sin β) = 2 – 2 cos (α – β)……(3)
Sum and Difference Formulas
Sum and Difference formulas of trigonometry are used to calculate the values of trigonometric functions at any angle where it is feasible to express the given angle as the sum or the difference of standard angles like 0°, 30°, 45°, 60°, 90°, and 180°. For example, to evaluate the value of the cosine function at 15°, we can write 15° as the difference between 45° and 30°; i.e., cos 15° = cos (45°-15°). Now with the help of sum and difference formulae, we can easily solve the above problem. In this article, we will learn about various Sum and Difference formulae used in trigonometry in detail.
Table of Content
- Trigonometry Functions
- What are Sum and Difference Formulas?
- Sum and Difference Formulae
- Proof of Sum and Difference Identities
- Sum and Difference Formulas for Cosine
- Cos (α – β) formula
- Cos (α + β) formula
- Sum and Difference Formulas for Sine
- Sin (α – β) formula
- Sin (α + β) formula
- Sum and Difference Formulas for Tangent
- Tan (α – β) formula
- Tan (α + β) formula
- Sum and Difference Formulae Table
- How to Apply Sum and Difference Formulas
- Solved Examples on Sum and Difference Formulas