Understanding the Trigonometric Expression

We are asked to find the value of the expression: $\cos 2A \cos 2B + \sin^2(A - B) - \sin^2(A + B)$. This problem requires us to use trigonometric identities to simplify the given expression.

Applying Trigonometric Identities

Let's break down the given expression:

The expression is: $\cos 2A \cos 2B + \sin^2(A - B) - \sin^2(A + B)$

We can focus on simplifying the part $\sin^2(A - B) - \sin^2(A + B)$. There is a standard trigonometric identity related to the difference of squares of sine functions:

The identity is: $\sin^2 x - \sin^2 y = \sin(x + y) \sin(x - y)$

Let's apply this identity to our expression. Here, let $x = A - B$ and $y = A + B$.

First, find $(x + y)$:

$x + y = (A - B) + (A + B) = A - B + A + B = 2A$

Next, find $(x - y)$:

$x - y = (A - B) - (A + B) = A - B - A - B = -2B$

Now, substitute these values into the identity $\sin^2 x - \sin^2 y = \sin(x + y) \sin(x - y)$:

$\sin^2(A - B) - \sin^2(A + B) = \sin(2A) \sin(-2B)$

Recall that $\sin(- \theta) = - \sin \theta$. So, $\sin(-2B) = - \sin 2B$.

Therefore, $\sin^2(A - B) - \sin^2(A + B) = \sin(2A) (- \sin 2B) = - \sin 2A \sin 2B$.

Simplifying the Complete Expression

Now substitute this simplified part back into the original expression:

Original expression: $\cos 2A \cos 2B + (\sin^2(A - B) - \sin^2(A + B))$

Substitute the simplified part:

$\cos 2A \cos 2B + (- \sin 2A \sin 2B) = \cos 2A \cos 2B - \sin 2A \sin 2B$

Now, we recognize another standard trigonometric identity for the cosine of the sum of two angles:

The identity is: $\cos(P + Q) = \cos P \cos Q - \sin P \sin Q$

Comparing our current expression, $\cos 2A \cos 2B - \sin 2A \sin 2B$, with this identity, we can see that it matches the right side if we let $P = 2A$ and $Q = 2B$.

Therefore, $\cos 2A \cos 2B - \sin 2A \sin 2B = \cos(2A + 2B)$.

Final Result and Conclusion

The simplified value of the given expression $\cos 2A \cos 2B + \sin^2(A - B) - \sin^2(A + B)$ is $\cos(2A + 2B)$.

Let's check this result against the given options.

• Option 1: $\sin (2A - 2B)$
• Option 2: $\sin (2A + 2B)$
• Option 3: $\cos (2A + 2B)$
• Option 4: $\cos (2A - 2B)$

Our simplified expression $\cos (2A + 2B)$ matches Option 3.

Revision Table: Key Trigonometric Identities

Here is a table summarizing the key identities used in this solution.

Additional Information: Exploring Related Concepts

Trigonometric identities are fundamental equations that are true for all values of the variables involved (where the expressions are defined). They are crucial for simplifying expressions, solving trigonometric equations, and proving other identities.

The identities used here are derived from the angle sum and difference formulas for sine and cosine:

• $\sin(x+y) = \sin x \cos y + \cos x \sin y$
• $\sin(x-y) = \sin x \cos y - \cos x \sin y$
• $\cos(x+y) = \cos x \cos y - \sin x \sin y$
• $\cos(x-y) = \cos x \cos y + \sin x \sin y$

From these, the product-to-sum and sum-to-product identities can also be derived, which are useful for simplifying various expressions.

For instance, the identity $\sin^2 x - \sin^2 y = \sin(x+y)\sin(x-y)$ can be derived as follows:

$\sin^2 x - \sin^2 y = (\sin x - \sin y)(\sin x + \sin y)$

Using sum-to-product formulas:

• $\sin x - \sin y = 2 \cos\left(\frac{x+y}{2}\right) \sin\left(\frac{x-y}{2}\right)$
• $\sin x + \sin y = 2 \sin\left(\frac{x+y}{2}\right) \cos\left(\frac{x-y}{2}\right)$

Multiplying these gives:

$(\sin x - \sin y)(\sin x + \sin y) = 4 \sin\left(\frac{x-y}{2}\right) \cos\left(\frac{x-y}{2}\right) \sin\left(\frac{x+y}{2}\right) \cos\left(\frac{x+y}{2}\right)$

Using the double angle identity $\sin(2\theta) = 2 \sin\theta \cos\theta$, we can rewrite this as:

$2 \sin\left(\frac{x-y}{2}\right) \cos\left(\frac{x-y}{2}\right) \times 2 \sin\left(\frac{x+y}{2}\right) \cos\left(\frac{x+y}{2}\right)$

$= \sin\left(2 \times \frac{x-y}{2}\right) \times \sin\left(2 \times \frac{x+y}{2}\right) = \sin(x-y) \sin(x+y)$.

Understanding these derivations helps in remembering and applying the identities correctly in problems involving trigonometric expressions.