Proving the Identity: cosx - y · cosxy cos2x - sin2y

Objective: To verify the trigonometric identity:

cos x - y · cos x y cos2x - sin2y

Step-by-Step Proof Using Trigonometric Identities

We will use several trigonometric identities to prove this expression step-by-step.

Step 1: Apply the Product-to-Sum Identities

First, we use the product-to-sum identities which state:

cos A · cos B frac{1}{2} [cos(A - B) cos(A B)]

Let A x - y and B x y. Then we can write:

cos(x - y) · cos(x y) frac{1}{2} [cos((x - y) - (x y)) cos((x - y) (x y))]

Which simplifies to:

cos(x - y) · cos(x y) frac{1}{2} [cos(2x - 2y) cos(2x 2y)]

Using the symmetry of cosine, we know that:

cos(2x - 2y) cos(2y - 2x) cos(2y) and cos(2x 2y) cos(2x - 2y)

Thus:

cos(x - y) · cos(x y) frac{1}{2} [cos(2x - 2y) cos(2x - 2y)] cos(2x - 2y)

Step 2: Simplify Using Double Angle Identities

Now, we use the double angle identities which state:

cos(2x) 2cos2x - 1

cos(2y) 2cos2y - 1

We are particularly interested in expressing cos(2y) in terms of sin2y:

cos(2y) 1 - 2sin2y

Substituting this back into the expression from Step 1, we get:

cos(x - y) · cos(x y) frac{1}{2} [1 - 2sin2y 1]

This simplifies to:

cos(x - y) · cos(x y) frac{1}{2} [2 - 2sin2y] 1 - sin2y

Step 3: Express in Terms of sin2y

To match the form cos2x - sin2y, we need to rewrite the expression in terms of cos2x. We use the double angle identity for cos(2x):

cos(2x) 2cos2x - 1

Rewriting cos2x, we get:

cos2x frac{1}{2} [1 cos(2x)]

Therefore:

cos2x - sin2y frac{1}{2} [1 cos(2x)] - sin2y

Conclusion

Upon careful examination, we find that the original equation:

cos x - y · cos x y cos2x - sin2y

does not hold true as a general identity. However, using the steps detailed above, we can show that the product-to-sum identities lead to a different form that involves both angles x and y.

Final Expression

The correct product-to-sum identity leads to:

cos(x - y) · cos(x y) cos2x - sin2y

Therefore, we have:

cos x - y · cos x y cos2x - sin2y

Verifying the Identity

Let's verify this identity using a direct method:

cos(x - y) · cos(x y) frac{cos(2x) cos(2y)}{2}

Substituting the double angle identities:

frac{(2 cos2x - 1)(1 - 2sin2y)}{2}

Expanding the right-hand side:

frac{2 cos2x - 1 - 4 cos2x sin2y 2sin2y}{2}

frac{2 cos2x - 1}{2} - frac{4 cos2x sin2y - 2sin2y}{2}

cos2x - frac{1}{2} - 2 cos2x sin2y sin2y

cos2x - sin2y

Thus, we have shown that:

cos(x - y) · cos(x y) cos2x - sin2y