Proving the Trigonometric Identity: cos^2(AB) - cos^2(A B) -sin(2A)sin(2B)

The world of trigonometry is rich with identities, each one serving a unique purpose in simplifying complex expressions and solving problems. One such identity, cos^2(AB) - cos^2(A B) -sin(2A)sin(2B), is a powerful tool in both theoretical and applied mathematics. In this article, we will rigorously prove this identity using fundamental trigonometric concepts and identities.

1. Understanding the Identity

The identity we are dealing with is cos^2(AB) - cos^2(A B) -sin(2A)sin(2B). Here, AB and A B are general angles, and the identity connects the cosine and sine functions in a specific manner.

2. Proving the Identity

To prove this identity, we will use a series of trigonometric conversions and simplifications. Let's begin with the left-hand side of the equation.

2.1 Step 1: Initial Transformation

Our starting point is the expression cos^2(AB) - cos^2(A B). We can use the difference of squares formula to rewrite this expression:

cos^2(AB) - cos^2(A B) [cos(AB) - cos(A B)][cos(AB) cos(A B)]

This step leverages the fact that x^2 - y^2 (x - y)(x y).

2.2 Step 2: Applying the Product-to-Sum Identities

Next, we apply the product-to-sum identities to the terms in the brackets:

cos(X) - cos(Y) -2sin((X Y)/2)sin((X - Y)/2)

cos(X) cos(Y) 2cos((X Y)/2)cos((X - Y)/2)

Applying these identities to our expression, we get:

[cos(AB) - cos(A B)] -2sin((AB A B)/2)sin((AB - A - B)/2)

[cos(AB) cos(A B)] 2cos((AB A B)/2)cos((AB - A - B)/2)

2.3 Step 3: Combining the Results

Now, let's combine these results:

-2sin((AB A B)/2)sin((AB - A - B)/2) * 2cos((AB A B)/2)cos((AB - A - B)/2)

Notice that we are multiplying two sine and two cosine terms. This can be simplified further:

-2 * 2 * sin((AB A B)/2) * sin((AB - A - B)/2) * cos((AB A B)/2) * cos((AB - A - B)/2) -4sin((AB A B)/2)cos((AB A B)/2)sin((AB - A - B)/2)cos((AB - A - B)/2)

2.4 Step 4: Using Double-Angle Identities

We can now use the double-angle identities, which state:

sin(2X) 2sin(X)cos(X)

Applying this to our expression:

-2 * sin((2AB 2A 2B)/4) * cos((2AB 2A 2B)/4) * sin((2AB - 2A - 2B)/4) * cos((2AB - 2A - 2B)/4) -2 * sin(2(AB A B)/4) * cos(2(AB A B)/4) * sin(2(AB - A - B)/4) * cos(2(AB - A - B)/4)

Simplifying further:

-2 * sin(AB A B) * cos(AB A B) * sin(AB - A - B) * cos(AB - A - B)

2.5 Step 5: Final Simplification

Finally, we can simplify the expression to:

-2 * sin(2A 2B)sin(2AB - 2B) -sin(2A 2B)sin(2A - 2B)

This is the required identity, verified as:

cos^2(AB) - cos^2(A B) -sin(2A)sin(2B)

3. Conclusion

In summary, we have shown that the identity cos^2(AB) - cos^2(A B) -sin(2A)sin(2B) holds true through a series of trigonometric manipulations and simplifications. This proof not only verifies the identity but also demonstrates the power of using fundamental trigonometric identities to solve complex problems.