Proving the Trigonometric Identity: cos^2(a) - sin^2(b) cos(ab) cos(a - b)

In the field of trigonometry, proving identities is an essential skill. One such notable identity is the expression:

cos^2(a) - sin^2(b) cos(ab) cos(a - b)

Understanding the Identity

This identity involves the manipulation of trigonometric functions with various angles. To prove this identity, we'll use the cosine addition and subtraction formulas.

Using the Cosine Addition and Subtraction Formulas

We start by recalling the cosine addition and subtraction formulas:

cos(ab) cos(a)cos(b) - sin(a)sin(b) cos(a - b) cos(a)cos(b) sin(a)sin(b)

By multiplying these two expressions, we get:

cos(ab) cos(a - b) [cos(a)cos(b) - sin(a)sin(b)][cos(a)cos(b) sin(a)sin(b)]

Simplifying the Expression

Using the difference of squares formula, we can simplify this expression:

cos(ab) cos(a - b) [cos(a)cos(b)]^2 - [sin(a)sin(b)]^2

Expanding both squares, we get:

cos(a)cos(b)^2 cos^2(a) cos^2(b)

sin(a)sin(b)^2 sin^2(a) sin^2(b)

Thus, we have:

cos(ab) cos(a - b) cos^2(a) cos^2(b) - sin^2(a) sin^2(b)

Relating Back to the Original Expression

To relate this back to our original expression, we express cos^2(a) in terms of 1 - sin^2(a):

cos^2(a) 1 - sin^2(a)

Substituting this back into our expression, we rewrite cos^2(a) cos^2(b) and sin^2(a) sin^2(b):

cos^2(a) cos^2(b) - sin^2(a) sin^2(b) (1 - sin^2(a)) cos^2(b) - sin^2(a) sin^2(b)

Expanding this, we get:

cos^2(b) - sin^2(a) cos^2(b) - sin^2(a) sin^2(b)

Since sin^2(b) cos^2(b) 1, we can simplify this to:

cos^2(b) - sin^2(a)

Conclusion

Thus, we can rewrite the expression:

cos^2(a) - sin^2(b) cos(ab) cos(a - b)

This completes the proof. Therefore, the identity is:

cos^2(a) - sin^2(b) cos(ab) cos(a - b)

Addendum: Alternative Proofs

Alternatively, the identity can also be proven by considering the right-hand side:

cos(ab) cos(a - b) [cos(a) cos(b) - sin(a) sin(b)][cos(a) cos(b) sin(a) sin(b)]

Expanding this, we get:

cos^2(a) cos^2(b) - sin^2(a) sin^2(b)

Expressing cos^2(a) as 1 - sin^2(a), we rewrite the expression:

(1 - sin^2(a)) cos^2(b) - sin^2(a) sin^2(b)

Expanding this, we get:

cos^2(b) - sin^2(a) cos^2(b) - sin^2(a) sin^2(b)

Cancelling out the terms, we are left with:

cos^2(b) - sin^2(a)

Therefore, the original equation holds true:

cos^2(a) - sin^2(b) cos(ab) cos(a - b)