Proving the Trigonometric Identity: cos(AB)cos(A-B) cos^2 A - sin^2 B
The trigonometric identity cos(AB)cos(A-B) cos^2 A - sin^2 B is a fundamental relationship in trigonometry. In this article, we will provide a detailed proof of this identity using basic trigonometric formulas.
Step-by-Step Proof of the Identity
Let's start by expressing both sides of the equation in a way that allows us to manipulate them using known trigonometric identities.
Left-Hand Side: cos(AB)cos(A-B)
Using the product-to-sum identities, we can rewrite cos(AB) and cos(A-B) as:
cos(AB) cosAcosB - sinAsinB cos(A-B) cosAcosB sinAsinBSubstituting these into the left-hand side:
cos(AB)cos(A-B) (cosAcosB - sinAsinB)(cosAcosB sinAsinB)
Expanding the product:
cos^2 A cos^2 B - sin^2 A sin^2 B
Now let's simplify the right-hand side:
Right-Hand Side: cos^2 A - sin^2 B
We start with the expression:
cos^2 A - sin^2 B
Using the Pythagorean identity, we know that:
sin^2 A 1 - cos^2 A sin^2 B 1 - cos^2 BSubstituting these identities, we get:
cos^2 A - (1 - cos^2 B)
Expanding this:
cos^2 A - 1 cos^2 B
Simplifying further:
cos^2 A cos^2 B - 1
Next, we need to show that the expanded left-hand side simplifies to the right-hand side:
cos^2 A cos^2 B - sin^2 A sin^2 B
Using the Pythagorean identity again:
cos^2 A cos^2 B - (1 - cos^2 A)(1 - cos^2 B)
Expanding the product:
cos^2 A cos^2 B - (1 - cos^2 A - cos^2 B cos^2 A cos^2 B)
Simplifying further:
cos^2 A cos^2 B - 1 cos^2 A cos^2 B - cos^2 A cos^2 B
Cancelling out the common terms:
cos^2 A cos^2 B - 1
Therefore, we have shown that:
cos(AB)cos(A-B) cos^2 A - sin^2 B
Conclusion
The identity cos(AB)cos(A-B) cos^2 A - sin^2 B is proven through a rigorous step-by-step approach using basic trigonometric identities. This identity is valuable in many areas of mathematics and physics, particularly in simplifying trigonometric expressions and solving complex trigonometric equations.