To prove the trigonometric identity \( \cos^2a - \sin^2b = \cos^2b - \sin^2a \), let's start by manipulating the given expression:

### Step 1: Start with the original equation
\[ \cos^2a - \sin^2b = \cos^2b - \sin^2a \]

### Step 2: Move terms around to one side of the equation
Let's move all terms to one side to see if we can simplify it:

\[ \cos^2a - \sin^2b - \cos^2b + \sin^2a = 0 \]

### Step 3: Group like terms
Now, group the cosine and sine terms together:

\[ (\cos^2a - \cos^2b) + (\sin^2a - \sin^2b) = 0 \]

### Step 4: Factor the expression
Notice that we can factor the grouped terms using the difference of squares identity:

\[ (\cos^2a - \cos^2b) + (\sin^2a - \sin^2b) = (\cos a - \cos b)(\cos a + \cos b) + (\sin a - \sin b)(\sin a + \sin b) = 0 \]

### Step 5: Analyze the equation
For the above equation to hold true for all angles \(a\) and \(b\), the following must be satisfied:

\[ (\cos a - \cos b)(\cos a + \cos b) = -(\sin a - \sin b)(\sin a + \

Learn how to prove the trigonometric identity cos^2a - sin^2b = cos^2b - sin^2a with detailed steps and explanations. This problem involves trigonometric concepts and the difference of squares identity. Suitable for high school students studying trigonometry.

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